3.1. Using Strong Typing to Error-Check Calculations
Linear algebra is an excellent setting for discussion of the
strong typing [
9] of a language, a concept used in the design of computer programming languages. The idea is that when the human-readable source code of a program is compiled (translated into machine-readable instructions), the compiler (the program which performs this translation) or runtime (the software which executes the code) verifies that the program objects are being used in a well-defined way, producing an error for each operation that is not well-defined. For example, a vector-type value would not be allowed to be added to a permutation-type value, even though tuples of unsigned integers (i.e., bytes) are used by the computer to represent both, and the computer’s processing unit could add together their byte-valued representations. However, such an operation would be meaningless with respect to the types of the operands. The result of the operation would depend on the non-canonical choice of representation for each object. Strong type checking has the advantage of catching many programming errors, including most importantly those resulting from an inherent misuse of the program’s objects. Within this paper, certain type-explicit notations will be used to provide forms of type awareness conducive to error-checking.
An important example of semi-strong typing in math is Penrose’s abstract index notation [
10], modeled on Einstein’s summation convention, in which linear algebra and tensor calculus are implemented using indexed objects (tensors) having a certain number and order of “up” and “down” indices (an abstraction of the genuine basis/coordinate expressions in which the indexed objects are arrays of scalars/functions). A non-indexed tensor is a scalar value, a tensor having a single up or down index is a vector or covector value respectively, a tensor having an up and a down index is an endomorphism, and so forth. The tensors are contracted by pairing a certain number of up indices with the same number of down indices, resulting in an object having as indices the uncontracted indices.
For example, given a finite-dimensional inner product space , where g is a -tensor (having the form , i.e., two down indices), a vector is a -tensor, and the length of v is . If , then has positive dimension, its vectors each being -tensors, and is an inner product on (which must be a -tensor in order to contract with two -tensors).
Certain type errors are detected by use of abstract index notation in the form of index mismatch. For example, with as above, if , then is a -tensor. Because of the repeated j down indices, the expression typically indicates a type error; cannot contract with because of incompatible valence (valence being the number of up and down indices). Furthermore, multiplying a -tensor with a -tensor without contraction should result in a -tensor, which should be denoted using three indices, as in .
The only explicit type information provided by abstract index notation is that of valence. The “semi” qualifier mentioned earlier is earned by the lack of distinction between the different spaces in which the tensors reside. For example, if are finite-dimensional vector spaces, then linear maps and can be written as -tensors, and their composition is written as the tensor contraction . However, while the expression makes sense in terms of valence compatibility (i.e., grammatically), the composition “” that it should represent is not well-defined. Thus this form of type error is not caught by abstract index notation, since the domains/codomains of the linear maps must be checked separately.
The use of dimensional analysis (the abstract use of units such as kilograms, seconds, etc.) in Physics is an important precedent of strong typing. Each quantity has an associated “dimension” (this is a different meaning from the “dimension” of linear algebra) which is expressed as a fraction of powers of formal symbols. The ordinary algebraic rules for fractions and formal symbols are used for the dimensional components, with the further requirement that addition and equality may only occur between quantities having the same dimension.
For example, if E, M and C represent the dimensions of energy, mass and cost, respectively, and if the energy storage density
of a battery manufacturing process is known (having dimensions energy per mass) and the manufacturing weight yield
of the battery is known (having dimensions mass per cost), then under the algebraic conventions of dimensional analysis, calculating the energy storage per cost (which should have dimensions energy per cost) is simple;
(the M symbols cancel in the fraction). Here, both
and
w are real numbers, and besides using the well-definedness of real multiplication, no type-checking is done in the expression
.
A contrasting example is the quantity , having dimensions . However, these dimensions may be considered to be meaningless in the given context. The quantity’s type adds meaning to the real-valued quantity, and while the quantity is well-defined as a real number, the uselessness of the type may indicate that an error has been made in the calculations. For example, a type mismatch between the two sides of an equation is a strong indication of error.
This is also a convenient way to think about the chain rule of calculus. If , , and x measure real-valued quantities, then measures the quantity z with respect to quantity x. Using Z, Y and X for the dimensions of the quantities z, y and x respectively, the derivative has units . When worked out, the dimensions for the quantities on either side of the equation will match exactly, having a non-coincidental similarity to the calculation in the battery product example.
3.3. Strongly-Typed Linear Algebra via Tensor Products
A fully strongly typed formulation of linear algebra will now be developed which enjoys a level of abstraction and flexibility similar to that of Penrose’s abstract index notation. Emphasis will be placed on notational and conceptual regularity via a tensor formalism, coupled with a notion of “untangled” expression which exploits and notationally depicts the associativity of linear composition.
If
V denotes a finite-dimensional vector space, then let
denote the
natural pairing on
V, and denote
using the infix notation
. The natural pairing is a nondegenerate bilinear form and its bilinearity gives the expression
multiplicative semantics (distributivity and commutativity with scalar multiplication), thereby justifying the use of the infix · operator normally reserved for multiplication. The natural pairing subscript
V is seemingly pedantic, but will prove to be an invaluable tool for articulating and navigating the rich type system of the linear algebraic and vector bundle constructions used in this paper. When clear from context, the subscript
V may be omitted.
Because
V is finite-dimensional, it is reflexive (i.e., the canonical injection
is a linear isomorphism). Thus the natural pairing
on
can be written naturally as
Note that . Though subtle, the distinction between and is important within the type system used in this paper.
Through a universal mapping property of multilinear maps, the bilinear forms
and
descend to the
natural trace maps
each extended linearly to non-simple tensors. These operations can also be called
tensor contraction. Noting that
and
are canonically isomorphic to
and
respectively, then for each
and
, it follows that
and
.
Definition 1 (Linear maps as tensors)
. Let V and W be finite-dimensional vector spaces, and let denote the space of vector space morphisms from V to W (i.e., linear maps). The linear isomorphism(extended linearly to general tensors) will play a central conceptual role in the calculations employed in this paper, as it will facilitate constructions which would otherwise be awkward or difficult to express. Linear maps and appropriately typed tensor products will be identified via this isomorphism. Given bases
and
, and dual bases
and
, a linear map
can be written under the identification in (1) as
where
, and in fact
is the matrix representation of
A with respect to the bases
and
, noting that the
i and
j indices denote the “output” and “input” components of
A respectively. Tensors are therefore the strongly typed analog of matrices, where the
type information is carried by the
component. One particular example is the the identity map on
V, which has type
and is expressed simply as
(equivalently,
). Throughout this paper, the identity map on
V will be referred to as the
identity tensor on V, or just the
identity tensor if
V is clear from context, and will be denoted as
.
One clarifying example of the tensor formulation is the adjoint operation of the natural pairing, also known as forming the dual of a linear map. It is straightforward to show that
(where the map is extended linearly to general tensors). This is literally the tensor abstraction of the matrix transpose operation; if
, then the dual
A is
. The matrix of
is precisely the transpose of the matrix of
A with respect to the relevant bases. The map * itself can be written as a 4-tensor
, where
.
There is a notion of the natural pairing of tensor products, which implements composition and evaluation of linear maps, and can be thought of as a natural generalization of scalar multiplication in a field. If
and
W are each finite-dimensional vector spaces, then the bilinear form
will be denoted also by the infix notation
(i.e.,
). If
V itself is a tensor product of
n factors which are clear from context, then
may be denoted by
(think an
n-fold tensor contraction). If
, then typically : is used in place of
. For example, from above,
.
Given a permutation , define a right-action by , mapping elements in the obvious way. For example, acting on puts the second factor in the third position, the third factor in the fourth position, and the fourth factor in the second, giving . This permutation is itself a linear map and of course can be written as a tensor. However, because it is defined in terms of a right action, the “domain factors” will come on the left. Thus is written as a tensor of the form (i.e., as a -tensor). Certain tensor constructions are conducive to using such permutations. In the above example, * can be written as .
The permutation right-action also works naturally when notated using superscripts. For example, if
, then
and so
When multiplying the permutations
and
in the third line, it is important to note that they are read left-to-right, since they are acting on
B on the right.
The inline cycle notation is somewhat ambiguous in isolation because the number of factors in the domain/codomain is not specified, let alone their types. This information can sometimes be inferred from context, such as from the natural pairing subscripts, as in the following examples.
Example 1 (Linearizing the inversion map)
. Let , i.e., the linear map inversion operator, where is an open submanifold of via the isomorphism . Its linearization (derivative) at in the direction is( is taken arbitrarily small due to the derivative being evaluated in an arbitrarily small neighborhood of ϵ = 0).In order to “move” the B parameter out so that it plays the same syntactical role as in the original expression , via adjacent natural pairing, some simple tensor manipulations can be done. The process is easily and accurately expressed via diagram. The following sequence of diagrams is a sequence of equalities. The diagram should be self-explanatory, but for reference, the number of boxes for a particular label denotes the rank of the tensor, with each box labeled with its type. The lines connecting various boxes are natural pairings, and the circles represent the unpaired “slots”, which comprise the type of the resulting expression.
The following step is nothing but moving the boxes for B out; the natural pairings still apply to the same slots, hence the cables dangling below.
In this setting, a tensor product amounts to flippantly gluing boxes together.
In order for B to be naturally paired in the same adjacent manner as in the original expression , the slots of must be permuted; the second moves to the third, the third to the fourth, and the fourth to the second.
The first diagram equals the last one, thus , and by the nondegeneracy of the natural pairing on , this implies that , noting that the statement of this expression does not require the direction vector B. The permutation exponent can be calculated easily using simple tensors, if not by the above diagrammatic manipulations;Here, the expression represents the expression . The next example will later be extended to the setting of Riemannian manifolds and their metric tensor fields, and put to use to formulate what are known as harmonic maps (see (3)). However, first, a new tensor operation must be defined.
Definition 2 (Parallel tensor product)
. If are vector spaces and and , then define their parallel tensor product byThe parentheses in the type specification are unnecessary, but hint at what the tensor decomposition for the quantity should be, if used as an operand to ⊠ again (see below). If A and B represent linear maps, then represents their tensor product as linear maps (the parentheses are unnecessary but hint at what the domain and codomain are, and for use of as an operand in another parallel tensor product), which is a “parallel” composition; if and , then .
There is a slight ambiguity in the notation coming from a lack of specification on how the tensor product of the operands is decomposed in the case when there is more than one such decomposition. Notation explicitly resolving this ambiguity will not be needed in this paper as the relevant tensor product is usually clear from context.
The parallel tensor product is associative; if
Y and
Z are also vector spaces and
, then
allowing multiply-parallel tensor products.
Example 2 (Tensor product of inner product spaces). If and are inner product spaces (noting that and are symmetric, i.e., literally invariant under ), then is an inner product space having induced inner product . Here, the “inputs” of A and B (the factors) are being paired using , while the “outputs” (the W factors) are being paired using , and the trace is used to “complete the cycle” by plugging the output into the input, thereby producing a real number. The expression can be written in a more natural way, which takes advantage of the linear composition, as (or, pedantically, ), instead of the more common but awkward trace expression mentioned earlier. In the tensor formalism, the inner product k should have type . Permuting the middle two components of the 4-tensor gives the correct type. In fact, . A further advantage to this formulation is that if any or all of are functions, there is a clear product rule for derivatives of the expression . This is something that is used critically in Riemannian geometry in the form of covariant derivatives of tensor fields (see (4)).
In this paper, the main use of the tensor formulation of linear maps is twofold: to facilitate linear algebraic constructions which would otherwise be difficult or awkward (this includes the ability to express derivatives of [possibly vector or manifold-valued] maps without needing to “plug in” the derivative’s directional argument), and to make clear the product-rule behavior of many important differentiable constructions.
3.4. Bundle Constructions
In order to use the calculus of variations involving Lagrangians depending on tangent maps of maps between smooth manifolds, it suffices to consider Lagrangians defined on smooth vector bundle morphisms. Continuing in the style of the previous section, a “full” tensor product of smooth vector bundles (4) will be formulated which will then allow expression of smooth vector bundle morphisms as tensor fields, sometimes called two-point tensor fields ([
11], p. 70). The full arsenal of tensor calculus can then be used to considerable advantage.
First, some definitions and simpler bundle constructions will be introduced. A smooth [fiber] bundle (hereafter referred to simply as a smooth bundle) is a 4-tuple where , E and N are smooth manifolds and is locally trivial, i.e., N is covered by open sets such that as smooth manifolds. The manifolds , E and N are called the typical fiber, the total space, and the base space respectively. The map is called the bundle projection. The full 4-tuple specifying a bundle can be recovered from the bundle projection map, so a locally trivial smooth map can be said to define a smooth bundle. The dimension of the typical fiber of a bundle will be called its rank, and will be denoted by or when the bundle is understood from context.
The
space of smooth sections of a smooth bundle defined by
is
and may also be denoted by
, if the bundle is clear from context. If nonempty,
is generally an infinite-dimensional manifold (the exception being when the base space
N is finite) [
12].
Proposition 1 (Trivial bundle)
. Let M and N be smooth manifolds. With anddefines a smooth bundle , called a trivial bundle.
Similarly, with and , is a trivial bundle. No proof is deemed necessary for (1), as each bundle projection trivializes globally in the obvious way. The symbol is a composite of × (indicating direct product) and → or ← (indicating the base space).
If
M and
N are smooth manifolds as in (1), then there are two particularly useful natural identifications.
These identifications can be thought of identifying a map
with its graph in
and
respectively. Furthermore, this allows bundle theory to be applied to reasoning about spaces of maps. The symbols
and
now carry a significant amount of meaning. Generally
will be used in this paper, for consistency with the
convention discussed in
Section 3.3. The symbols
and
are examples of telescoping notation, as they are built notationally on ×, and conceptually on the direct product, which is what is denoted by ×. The arrow portion of the symbols can be discarded when type-specificity is not needed.
Proposition 2 (Direct product bundle)
. Let and be smooth bundles. Thendefines a smooth bundle . This bundle is called the direct product of and , and is not necessarily a trivial bundle. Proof. Let
and
trivialize
and
over open sets
and
respectively. Then
has inverse
. Note that
and that
defines a diffeomorphism. Then
defines a diffeomorphism, and
showing that
trivializes
over
. Since
can be covered by such trivializing sets, this establishes that
defines a smooth bundle. The typical fiber of
is
. □
A
smooth vector bundle is a fiber bundle whose typical fiber is a vector space and whose local trivializations are linear isomorphisms when restricted to each fiber. If
is a smooth vector bundle, then its
dual vector bundle is a smooth vector bundle defined in the following way.
Because
is a vector space, the notation
is already defined. In analogy with
Section 3.3, there are natural pairings on a vector bundle and its dual, defined simply by evaluation. If
,
and
, then
and
. Both expressions evaluate to
. Natural traces and
n-fold tensor contraction can be defined analogously. Again, while seemingly pedantic, the subscripted natural pairing notation will prove to be a valuable tool in articulating and error-checking calculations involving vector bundles. To generalize the rest of
Section 3.3 will require the definition of additional structures.
For the remainder of this section, let and now be smooth vector bundles. The following construction is essentially an alternate notation for , but is one that takes advantage of the fact that and are vector bundles, and encodes in the notation the fact that the resulting construction is also a vector bundle. This is analogous to how is a vector space with a natural structure if V and W are vector spaces, except that this is usually denoted by .
Proposition 3 (“Full” direct sum vector bundle)
. IfThendefines a smooth vector bundle , called the full direct sum of and .
For each , the vector space structure on is given in the following way. Let and . Then It is critical to see (1) for remarks on notation.
Proof. Let
U,
V,
,
,
P,
,
and
be as in the proof of (2), and define
. Noting that
is a smooth bundle isomorphism over
, so to show that
is a linear isomorphism in each fiber, it suffices to show that it is linear in each fiber. Let
,
and
. Then
Thus
is linear in each fiber, and because it is invertible, it is a linear isomorphism in each fiber. In particular,
is a smooth vector bundle isomorphism over
. Applying
to the above equation gives
as desired. □
This construction differs from the Whitney sum of two vector bundles, as the base spaces of the bundles are kept separate, and are not even required to be the same. This allows the identification of as , which may be done without comment later in this paper. Some important related structures are and , where .
The next construction is what will be used in the implementation of smooth vector bundle morphisms as tensor fields.
Proposition 4 (“Full” tensor product bundle)
. IfThendefines a smooth vector bundle , called the full tensor product (This construction is alluded to in ([13], p. 121), but is not defined or discussed.) of and .
It is critical to see (1) for remarks on notation.
Proof. Since the argument in the definition of is not necessarily unique, the well-definedness of must be shown. Let . Then in particular, for some , and therefore and for each index i and j. Thus and , so the expression defining is well-defined.
The set does not have an a priori global smooth manifold structure, as it is defined as the disjoint union of vector spaces. A smooth manifold structure compatible with that of the constituent vector spaces will now be defined.
Let
and
trivialize
and
over open sets
and
respectively, such that
and
are each linear in each fiber. Define
with element mapping
The map
is well-defined and smooth in each fiber by construction, since for each
,
is a linear isomorphism by construction. Additionally,
has been constructed so that
on
. Define the smooth structure on
by declaring
to be a diffeomorphism. The map
is trivialized over
. The set
can be covered by such trivializing open sets. Thus
has been shown to be locally diffeomorphic to the direct product of smooth manifolds, and therefore it has been shown to be a smooth manifold. With respect to the smooth structure on
, the map
is smooth, and has therefore been shown to define a smooth vector bundle. □
Remark 1 (Notation regarding base space). The “full” direct sum (3) and “full” tensor product (4) bundle constructions allow direct sums and tensor products to be taken of vector bundles when the base spaces differ. If the base spaces are the same, then the construction “joins” them, producing a vector bundle over that shared base space. For example, if E and F are vector bundles over M, then has base space , while has base space M. The base space can be specified in either case as a notational aide; the latter example would be written as . If no subscript is provided on the ⊗ symbol, then the base spaces are “joined” if possible (if they are the same space), otherwise they are kept separate, as in the “full” tensor product construction. This notational convention conforms to the standard Whitney sum and tensor product bundle notation, and uses the notion of telescoping notation to provide more specificity when necessary.
Given a fiber bundle, a natural vector bundle can be constructed “on top” of it, essentially quantifying the variations of bundle elements along each fiber. This is known as the vertical [tangent] bundle ([
12], p. 43), and it plays a critical role in the development of Ehresmann connections, which provide the “horizontal complement” to the vertical bundle.
Proposition 5 (Vertical bundle). Let define a smooth [fiber] bundle. If , then defines a smooth vector bundle subbundle of , called the vertical bundle over E. Furthermore, the fiber over is .
Proof. Because is a smooth surjective submersion, is a subbundle of having corank and therefore rank equal to that of E. Furthermore, if and , then represents an arbitrary element of , and , showing that , and therefore that . This shows that . Because , this shows that . □
Given the extra structure that a vector bundle provides over a [fiber] bundle, there is a canonical smooth vector bundle isomorphism which adds significant value to the pullback bundle formalism used throughout this paper. This can be seen put to greatest use in
Section 4, for example, in development of the first variation (see (1)).
Proposition 6 (Vertical bundle as pullback)
. If defines a smooth vector bundle, thenis a smooth vector bundle isomorphism over , called the vertical lift, having inversewhere, without loss of generality, is an E-valued variation which lies entirely in a single fiber. Proof. It is clear that is linear and injective on each fiber. By a dimension counting argument, it is therefore an isomorphism on each fiber. Because it preserves the basepoint, it is a vector bundle isomorphism over . Because the map is smooth, so is the defining expression for , thereby establishing smoothness. That inverts is a trivial calculation. □
3.6. Pullback Bundles
The pullback bundle, defined below, is a crucial building block for many important bundle constructions, as it enriches the type system dramatically, and allows the tensor formulation of linear algebra to be extended to the vector bundle setting. In particular, the abstract, global formulation of the space of smooth vector bundle morphisms over a map is achieved quite cleanly using a pullback bundle. Furthermore, the use of pullback bundles and pullback covariant derivatives simplifies what would otherwise be local coordinate calculations, thereby giving more insight into the geometric structure of the problem.
For the duration of this section, let be a smooth bundle having rank r.
Proposition 7 (Pullback bundle)
. Let M and N be smooth manifolds and let be smooth. Ifandthen defines a smooth bundle. In particular, is a smooth manifold having dimension . The bundle defined by is called the pullback of by .
Proof. Recalling that
denotes the typical fiber of
, let
trivialize
over open set
. Define
and
Claim (1): and are smooth. Proof: , and is clearly smooth as a map defined on the larger manifold. Therefore it restricts to a smooth map on . An analogous argument shows that is smooth. Claim (1) proved.
Claim (2):
inverts
. Proof: Let
. Then
With
,
proving Claim (2).
Claim (3):
trivializes
over
. Proof: Let
. Then
and by claims (1) and (2),
is a diffeomorphism, so
trivializes
over
. Claim (3) proved.
Since M can be covered with sets as in claim (3) and since the typical fiber of is diffeomorphic to , this shows that defines a smooth bundle . Because is locally diffeomorphic to the product of an open subset of M with , has been shown to be a smooth manifold having dimension . □
While the pullback bundle is constructed as a submanifold of a direct product, there is a natural bundle morphism into the pulled-back bundle, which serves as an interface to maps defined on the pulled-back bundle. Usually this morphism is notationally suppressed, just as naturally isomorphic spaces can be identified without explicit notation.
Corollary 1 (Pullback fiber projection bundle morphism)
. If is smooth, thenis a smooth bundle morphism over ϕ which is an isomorphism when restricted to any fiber of . Because is the projection , its tangent map is also just the projection .
Proposition 8 (Bundle pullback is a contravariant functor)
. The map of categoriesis a contravariant functor. Here, naturally isomorphic bundles in , for each manifold M, are identified (along with the corresponding morphisms). Proof. Noting that
and that
it follows that
, i.e.,
satisfies the identity axiom of functoriality.
For the contravariance axiom, let
and
be smooth manifold morphisms and let
be a smooth bundle. Then
and
showing that
, and therefore
establishing
as a contravariant functor. □
The space of sections of a pullback bundle is easily quantified.
This space will be central in the theory developed in the rest of this paper. Furthermore, it is naturally identified with the space of sections along the pullback map;
These spaces are naturally isomorphic to one another, and therefore an identification can be made when convenient. While the former space is more correct from a strongly typed standpoint, the latter space is a convenient and intuitive representational form. The particular correspondence depends heavily on the fact that
is a submanifold of
.
Furthermore, if
, then
. Note that it is
not true that any
can be written as
for some
, for example when there exists some distinct
such that
and
. Furthermore, the representation
is generally non-unique, for example when
is not surjective, sections
which differ only away from the image of
will still give
. Before developing the notion of a linear connection on a pullback bundle, it will be necessary to address these features which, while inconvenient, provide the strength of the pullback bundle and pullback covariant derivative (see (5)).
Lemma 1 (Local representation of elements). Recall that r denotes the rank of smooth bundle F. If then each point has some neighborhood U in which σ can be written locally as , where is a frame for , and are defined by .
Proof. Let
, let
be a neighborhood of
over which
is trivial, and let
, so that
U is a neighborhood of
p. Let
be a frame for
(i.e.,
), and let
be the corresponding coframe (i.e., the unique
such that
for each
). Define
by
. Then
as desired. □
Some literature uses expressions of the form along with an implicit use of the section-identifying isomorphism to write down particular sections of pullback bundles. In most cases, this tacit identification of spaces is harmless, but certain highly involved calculations may suffer from it. The section that corresponds to under said isomorphism is . However, because this expression is unwieldy and therefore a more compact and contextually meaningful expression is called for.
Definition 3 (Pullback section)
. If and is smooth, then defineThis is known as a pullback section.
The pullback section is deservedly named. If and are smooth, then in the sense of the proof of (8).
Proposition 9 (Bundle pullback commutes with tensor product)
. If E and F are smooth vector bundles over manifold N and is smooth, then the map(extended linearly to general tensors) is a smooth vector bundle isomorphism. Proof. Let c denote the above map. The well-definedness of c comes from the universal mapping property on multilinear forms which induces a linear map on a corresponding tensor product. If , then which implies that or , and therefore that . Because there exists a basis for consisting only of simple tensors, this implies that c is injective, and by a dimensionality argument, that c is an isomorphism. The map is clearly smooth and respects the fiber structures of its domain and codomain. Thus c is a smooth vector bundle isomorphism. □
The contravariance of pullback and its naturality with respect to tensor product are two essential properties which provide some of the flexibility and precision of the strongly typed tensor formalism described in this paper. This will become quite apparent in
Section 4.
Remark 2 (Tensor field formulation of smooth vector bundle morphisms)
. A particularly useful application of pullback bundles is in forming a rich-type system for smooth vector bundle morphisms. This approach was inspired by ([14], p. 11). Let and be smooth vector bundles, and let be smooth. Consider , i.e., the space of smooth vector bundle morphisms over the map ϕ. There is a natural identification with another space which lets the base map ϕ play a more direct role in the space’s type. In particular,This particular identification of smooth vector bundle morphisms over ϕ can now be directly translated into the tensor field formalism, analogously to (1).The inverse image of is given locally; let and denote local frames for E and F in neighborhoods and respectively, with , and let and denote their dual coframes. Then the tensor field corresponding to B is given locally in U by , where .Quantifying smooth vector bundle morphisms as tensor fields lends itself naturally to doing calculus on vector and tensor bundles, as the relevant derivatives (covariant derivatives) take the form of tensor fields. The type information for a particular vector bundle morphism is encoded in the relevant tensor bundle.
3.7. Tangent Map as a Tensor Field
This section deals specifically with the tangent map operator by using concepts from
Section 3.5 and
Section 3.6 to place it in a strongly typed setting and to prepare to unify a few seemingly disparate concepts and notation for some tangible benefit (in particular, see
Section 3.10).
Given a smooth map
, its tangent map
is a smooth vector bundle morphism over
, so by (2), is naturally identified with a tensor field
which may be denoted by
where type pedantry is deemed unnecessary. This construction is known as a
two-point tensor field ([
11], p. 70). In general, if
, then
and
have distinct types
and
respectively, and therefore have no well-defined sum. Thus
is a nonlinear derivative. The inscribed ∘ symbol within the symbol
is meant to denote that nonlinearity, in particular distinguishing it from a linear covariant derivative.
Remark 3 (Generalized covariant derivative)
. The well-known one-to-one correspondence between linear connections and linear covariant derivatives ([6], p. 520) generalizes to a one-to-one correspondence between Ehresmann connections and a generalized notion of covariant derivative. To give a partial definition for the purposes of utility, a generalized covariant derivative on a smooth [fiber] bundle is a map ∇ on such that for each . The space of maps is naturally identified as , and there is a natural Ehresmann connection on the bundle , whose corresponding covariant derivative is the tangent map operator. This is the subject of another of the author’s papers and will not be discussed here further. This is mentioned here to incorporate linear covariant derivatives (to be introduced and discussed in Section 3.8) and the tangent map operator (a nonlinear covariant derivative) under the single category “covariant derivative”. There is a subtle issue regarding construction of the cotangent map of which is handled easily by the tensor field construction. In particular, while the cotangent map is the pointwise adjoint of the tangent map , i.e., for each , is linear and is the adjoint of , it does not follow that , being some sort of “total adjoint” of . The obstruction is due to the fact that may not be surjective, so there may be some fiber that is not of the form , and therefore the domain could not be all of . Furthermore, even if were surjective, if it were not also injective, say for some distinct , then , and , so the action on the fiber is not well-defined.
In the tensor field parlance, the cotangent map
simply takes the form
The permutation superscript
is used here instead of * to distinguish it notationally from pullback notation, which will be necessary in later calculations. The key concept is that the tensor field
encodes the base map
; the basepoint
is part of the domain
itself.
The chain rule in the tensor field formalism makes use of the bundle pullback. If
is smooth, then
Because
, to form a well-defined natural pairing, the use of the pullback
is necessary (instead of just
).
Sometimes it is useful to discard some type information and write
i.e.,
such that
. This is easily done by the canonical fiber projection available to all pullback bundle constructions;
, and the canonical fiber projection is
as defined in (1). The granularity of the type system should reflect the weight of the calculations being performed. For demonstration of contrasting situations, see the discussion at the beginning of
Section 3.8 and the computation of the first variation in (1).
It is important to have notation which makes the distinction between the smooth vector bundle morphism formalism and the tensor field formalism, because it may sometimes be necessary to mix the two, though this paper will not need this. An added benefit to the tensor field formulation of tangent maps is that certain notions regarding derivatives can be conceptually and notationally combined, for example in
Section 3.10.
3.8. Linear Covariant Derivatives
As will be shown in the following discussion, a linear covariant derivative (commonly referred to in the standard literature without the “linear” qualifier) provides a way to generalize the notion in elementary calculus of the differential of a vector-valued function. The linear covariant derivative interacts naturally with the notion of the pullback bundle, and this interaction leads naturally to what could be called a covariant derivative chain rule, which provides a crucial tool for the tensor calculus computations seen later.
Let
V and
W be finite-dimensional vector spaces let
be open, and let
be differentiable. Recall from elementary calculus the differential
(essentially matrix-valued). There is no base map information encoded in
(i.e.,
cannot be recovered from
alone), it contains only derivative information. The vector space structure of
V and
W allows the trivializations
and
, where the first factors are the base spaces and the second factors are the fibers (see (1)). The tangent map
(see
Section 3.7) has a codomain that can be trivialized similarly;
Because
, as a set, is a direct product, it can be decomposed into two factors. Letting
and
be the projections onto the first and second factors respectively,
The map
is the element of
identified with the base map
itself;
. This base map information is discarded in defining the differential of
as
; the fiber portion of
. This construction relies critically on the natural isomorphism
for a vector space
W.
An analogous construction shows that the differential of a map is well-defined even when its domain is a manifold. However, when the codomain of a map is only a manifold, there does not in general exist a natural trivialization of its tangent bundle (in contrast to the vector space case), and therefore cannot be defined without additional structure. A linear covariant derivative provides the missing structure.
For the remainder of this section, let define a smooth vector bundle having rank r.
A linear covariant derivative on E provides a means of taking derivatives of sections of E (i.e., maps such that ) without passing to a higher tangent bundle as would happen under the tangent map functor (i.e., if then and ). A linear covariant derivative provides an effective “trivialization” of analogous to the trivialization as discussed above, discarding all but the “fiber” portion of the derivative, allowing the construction of an object known as the total linear covariant derivative analogous to the differential as discussed above.
The notion of a linear covariant derivative on a vector bundle is arguably the crucial element of differential geometry (The
Fundamental Lemma of Riemannian Geometry establishes the existence of the Levi-Civita connection ([
8], p. 68), which is a linear covariant derivative satisfying certain naturality properties.). In particular, this operator implements the product rule property common to anything that can be called a derivation—a property which is particularly conducive to the operation of tensor calculus. The total linear covariant derivative of a vector field (i.e., section of a vector bundle) allows the generalization of many constructions in elementary calculus to the setting of smooth vector bundles equipped with linear covariant derivatives. For example, the divergence
of a vector field
X on
generalizes to the divergence
of a vector field
X on
N, which has an analogous divergence theorem among other qualitative similarities.
Remark 4 (Natural linear covariant derivative on trivial line bundle)
. Before making the general definition for the linear covariant derivative, a natural linear covariant derivative will be introduced. With N denoting a smooth manifold as before, if , then is the differential of f. LetBecause is naturally identified with , this is essentially the natural linear covariant derivative on the trivial line bundle . Note that there is an associated product rule; if , then , andWhen clear from context, the superscript decoration can be omitted and the derivative denoted as . Definition 4 (Linear covariant derivative)
. A linear covariant derivative on a vector bundle defined by is an -linear map satisfying the product rulewhere and . The switch in order in the first term of the expression is necessary to form a tensor field of the correct type, . If , then the expression is known as the total [linear] covariant derivative of σ. If [in a subset ], then σ is said to be parallel [on U]. The “linear” qualifier is implied in standard literature and is therefore often omitted. The inscribed | in is to indicate that the covariant derivative is linear, and can be omitted when clear from context, or when it is unnecessary to distinguish it from the nonlinear tangent map operator whose decorated symbol is . For the remainder of this section, this distinction will not be necessary, so an undecorated ∇ will be used.
For
, it is customary to denote
by
, where
V indicates the “directional” component of the derivative. Following this convention, the product rule can be written in a form where the product rule is more obvious;
Given a linear covariant derivative
on
E, there is a naturally induced linear covariant derivative
on
satisfying the product rule for the natural pairing on
E, namely,
where
,
, and
.
A covariant derivative is a local operator with respect to the base space
N; if
, then
depends only on the restriction of
to an arbitrarily small neighborhood of
p ([
8], p. 50), and therefore the restriction
makes sense, allowing calculations using local expressions. Furthermore, a covariant derivative can be constructed locally and glued together under certain conditions. See ([
6], p. 503) for more on this, and as a reference for general theory on bundles, covariant derivatives, and connections.
Linear covariant derivatives on several vector bundle constructions will now be developed. In analogy to defining a linear map by its action on a generating subset (e.g., a basis or a dense subspace) and then extending using the linear structure, Lemma (3) allows a covariant derivative to be defined on a generating subset (which can be chosen to make the defining expression particularly natural) and then extending. In this case, the relevant space is the space of sections of the vector bundle, which is a module over the ring of smooth functions on a manifold, and the extension process is done via linearity and the product rule (see (4)). This approach will allow the local trivialization implementation details to be hidden within the proof of Lemma (3)—an example of information hiding—so that constructions of covariant derivatives can proceed clearly by focusing only on the natural properties of the relevant objects and then invoking the lemma to do the “dirty” work (see (10) and (11)).
A bit of useful notation will be introduced to simplify the next definition. If is a subset of a -module whose elements are functions on N (and therefore have a notion of restriction to a subset) and is open, then let denote the set of restrictions of the elements of G to the set U. Note that by construction.
Definition 5 (Finitely generating subset). Say that a subset of a module finitely generates the module if the subset contains a finite set of generators for the module.
Definition 6 (Locally finitely generating subset). If Γ is a -module and , then G is said to be a locally finitely generating subset of if each point has a neighborhood for which finitely generates .
The space of sections of a vector bundle is the archetype for the above definition. The locally trivial nature of allows local frames to be chosen in a neighborhood of each point of N, from which global smooth sections (though not necessarily a global frame) can be made using a partition of unity subordinate to the trivializing neighborhoods. The set of such global sections forms a locally finite generating subset of .
Lemma 2. If G is a locally finitely generating subset of , then each point in N has a neighborhood and such that forms a frame for . In other words, a local frame can be chosen out of G near each point in N.
Proof. Let and let be a neighborhood of q for which finitely generates (here, , recalling that ). Without loss of generality, let be linearly independent (this is possible because spans the vector space ). Because is continuous for each i and the linear independence of the sections is an open condition (defined by where ), there is a neighborhood of q for which is a linearly independent set for each . Finally, letting for , the sections form a frame for . □
The following lemma shows that defining a covariant derivative on a locally finitely generating subset of the space of sections of a vector bundle is sufficient to uniquely define a covariant derivative on the whole space. The particular generating subset can be chosen so the covariant derivative has a particularly natural expression within that subset.
Lemma 3 (Linear covariant derivative construction). Let G be a locally finite generating subset of . If satisfies the linear covariant derivative axioms (What is meant by this is that the product rule must only be satisfied on if , where and .), then there is a unique linear covariant derivative whose restriction to G is .
Proof. If
, then by (2) there exists a neighborhood
of
q for which there are
forming a frame for
. If
, then
for some
(specifically,
, where
denotes the dual coframe of
). Define
locally on
so as to satisfy the product rule
To show well-definedness, let
be another frame for
. Then
for some
. Let
be the unique smooth vector bundle isomorphism such that
. Writing
and
with respect to the frame
as
and
respectively, it follows that
and
. Then
The last equality follows because
, which is a constant function, so
Thus the expression defining
does not depend on the choice of local frame. This establishes the well-definedness of
.
Clearly the restriction of to G is . This establishes the claim of existence. Uniqueness follows from the fact that is defined in terms of the maps and . □
Lemma (3) is used in the proof of the following proposition to allow a natural formulation of the pullback covariant derivative with respect to a natural locally finite generating subset of , in which the relevant derivative has a natural chain rule.
Proposition 10 (Pullback covariant derivative)
. If is smooth and is a covariant derivative on E, then there is a unique covariant derivative on satisfying the chain rulefor all . Proof. Let
noting that a local frame
over open set
induces a local frame
, so
G is a locally finite generating subset of
. Define
The well-definedness and
-linearity of
comes from that of
. For the product rule, if
and
, then the product
is an element of
G if and only if
for some
, in which case,
. Then it follows that
which is exactly the required product rule. By (3), there exists a unique covariant derivative
on
whose restriction to
G is
. □
The full notation is often cumbersome, so it may be denoted by when the pulled-back bundle is clear from context.
Remark 5. There is an important feature of a pullback covariant derivative in the case that pullback map is not an immersion; the pullback covariant derivative may be nonzero even where the pullback map is singular. This fact can be obscured by a certain abuse of notation which often comes in the expression of the geodesic equations in differential geometry (see (4)). An example will illustrate this point.
Let be a covariant derivative on . Let be a unit-length vector field which describes the location of a person (the basepoint) and direction s/he is looking (the fiber portion) with respect to time (let have standard coordinate t). Define by , so that θ is the base map of Θ, i.e., θ has discarded the direction information and only encodes the location information. Say that for some closed interval , is identically zero (and so is not an immersion), but that is nonvanishing; see Figure 1. Mathematically, this means that during this time, Θ is varying only within a single fiber of . Physically, this means that during this time, the person is standing still but the direction s/he is looking is changing. Passing to a higher tangent space is often undesirable (note that takes values in ), so to avoid this, a covariant derivative is used. In order to be meaningful, the covariant derivative must capture this fiber-only variation. Because Θ is a vector field along θ, it can be written as , and the covariant derivative on induces a pullback covariant derivative on , which has base space . In other words, is parameterized by time. Then is the desired covariant derivative of Θ with respect to time. A coordinate-based calculation will be made to make completely obvious why this pullback covariant derivative captures the desired information. Let be local coordinates on M and, for simplicity, assume that the image of θ lies entirely within this coordinate chart. Because is a local frame for , is a local frame for , by (1) and can be written locally as for some functions . ThenNote that . Within the interval I, vanishes, so the second term vanishes on I. However, because Θ is varying in a fiber-only direction within I, the basepoint is not changing and can be identified with an elementary vector space derivative (the fiber is a vector space and so an elementary derivative is well-defined there). This fiber-direction derivative is nonvanishing by assumption, so is nonvanishing on I as desired. Introducing a bit of natural notation which will be helpful for the next result, if
and
, then define
and
by
for each
.
Proposition 11 (Induced covariant derivatives on
and
)
. If and are covariant derivatives on E and F respectively, then there are unique covariant derivativesandon and respectively, satisfying the sum ruleand the product rulerespectively, where , , and . Here, (and its dual) is used instead of the isomorphic vector bundle (and its dual). Proof. Suppressing the pedantic use of the
subscript to avoid unnecessary notational overload, the set
is a locally finite generator of
, since local frames for
take the form
, where
and
are local frames for
E and
F respectively. Define
This map is well-defined and
-linear by construction, since the connections
and
are well-defined and
-linear. If
,
, and
, then the product
is in
G (i.e., has the form
for some
and
) if and only if
is constant. Thus the product rule (restricted to elements of
G) reduces to
-linearity, which is already satisfied. By (3), there exists a unique connection
on
whose restriction to
G is
.
Similarly, the set
is a locally finite generator of
, since local frames for
take the form
, where
and
are local frames for
E and
F respectively. Define
This map is well-defined and
-linear by construction, since the connections
and
are well-defined and
-linear. For the product rule, with
,
, and
, the product
is in
H if and only if there exist
and
such that
(noting that then
). In this case, with
,
which is exactly the required product rule. By (3), there exists a unique connection
on
whose restriction to
H is
. □
Remark 6 (Naturality of the covariant derivatives on
and
).
Letting () for brevity, the mapsandeach extended linearly to the rest of their domains, are easily shown to be smooth vector bundle isomorphisms over . Thenandfor all , , and , showing that the connections on and are ξ and ψ-related to the naturally induced connections on and respectively, and are therefore in this sense natural. The sum and product correspond to and under ξ and ψ respectively. Many important tensor constructions involve permutations. An extremely useful property of these permutations is that they commute with the covariant derivatives induced by the covariant derivatives on the tensor bundle factors, making them natural operators in the setting of covariant tensor calculus.
Proposition 12 (Transposition tensor fields are parallel). Let be smooth vector bundles over M having covariant derivatives respectively, let and , and let and denote the induced covariant derivatives.
If denotes the tensor field which maps to (i.e., transposes the second and third factors), then is a parallel tensor field with respect to the covariant derivative induced on the vector bundle , i.e., .
Proof. Let
. Then
Because
X is arbitrary, this shows that
. This extends linearly to general tensors, so
, as desired. □
The fact that all transposition tensor fields are parallel implies that all permutation tensor fields are parallel, since every permutation is just the product of transpositions. This gives as an easy corollary that a covariant derivative operation commutes with a permutation operation, which has quite a succinct statement using the permutation superscript notation.
Corollary 2 (Permutation tensor fields are parallel)
. Let be smooth vector bundles over M each having a covariant derivative, and let and . If is interpreted as the tensor field in which maps to , then σ is a parallel tensor field. Stated using the superscript notation, with and , Proof. This follows from the fact that
can be written as the product of transpositions;
because of the product rule and because each transposition is parallel. The claim regarding commutation with the superscript permutation follows easily from its definition.
using the fact that
, since
is a parallel tensor field. □
Finally, any smooth vector bundle has a canonical identity tensor field acting as the identity on , i.e., for all . Given a local frame over open set , it has the expression . The identity tensor field is an invaluable tool in forming tensor field expressions and in phrasing other naturality conditions regarding covariant derivatives.
Proposition 13 (Identity tensor field is parallel). Let be a smooth vector bundle with a linear covariant derivative . Then is parallel with respect to , i.e., .
Proof. Let
. Then by definition,
. Taking the covariant derivative of both sides with respect to
,
Because
is arbitrary, this implies that
. Because
X is arbitrary, this implies that
. □
3.9. Decomposition of
In using the calculus of variations on a manifold
M where the Lagrangian is a function of
(this form of Lagrangian is ubiquitous in mechanics), taking the first variation involves passing to
. Without a way to decompose variations into more tractable components, the standard integration-by-parts trick ([
15], p. 16) cannot be applied. The notion of a local trivialization of
via choice of coordinates on
M is one way to provide such a decomposition. A coordinate chart
on
M establishes a locally trivializing diffeomorphism
. However such a trivialization imposes an artificial additive structure on
depending on the [non-canonical] choice of coordinates, only gives a local formulation of the relevant objects, and the ensuing coordinate calculations do not give clear insight into the geometric structure of the problem. The notion of the linear connection remedies this ([
16]).
A
linear connection on the vector bundle
is a subbundle
of
such that
and
for all
and
, where
is the scalar multiplication action of
a on
E ([
6], p. 512). The bundle
may also be called a
horizontal space of the vector bundle
(“a” is used instead of “the” because a choice of
is generally non-unique). For convenience, define
noting then that
.
A linear connection can equivalently be specified by what is known as a connection map; essentially a projection onto the vertical bundle. This is a slightly more active formulation than just the specification of a horizontal space, as a covariant derivative can be defined directly in terms of the connection map—see ([
6], p. 518), ([
17], p. 128), ([
18], p. 173), and ([
7], p. 208).
Proposition 14 (Connection map formulation of a linear connection)
. If (i.e., is a smooth vector bundle morphism over π) is a left-inverse for that is equivariant with respect to and (i.e., ) ([7], p. 245), then defines a linear connection on the vector bundle . Such a map v is called the connection map associated to H. Conversely, given a linear connection H, there is exactly one connection map defining H in the stated sense. Proof. That v is a left-inverse for implies that v has full rank, so defines a subbundle of having the same rank as . Because v is smooth, H is a smooth subbundle. Furthermore, the condition implies that for each , and therefore by a rank-counting argument.
If and , then , which equals zero if and only if , i.e., if and only if . Thus . This establishes as a linear connection.
Conversely, if H is a linear connection and and are connection maps for H, then . Then because the image of is all of , it follows that . Since by definition, and since , this shows that . Uniqueness of connection maps has been established. To show existence, define , where be the canonical projection, recalling that . It is easily shown that v is a connection map for H. □
Proposition 15 (Decomposing
)
. If is a connection map, thenis a smooth vector bundle isomorphism over . See Figure 2. Proof. Because
, and
and
, the fiber-wise restriction
is a linear isomorphism for each
. The map is a smooth vector bundle morphism over
by construction. It is therefore a smooth vector bundle isomorphism over
. □
Remark 7 (Linear connection/covariant derivative correspondence)
. Given a covariant derivative on a smooth vector bundle , there is a naturally induced linear connection, defined via the connection mapwhere is a variation of . Here, denotes the pullback of the covariant derivative through the map (see (10)). Conceptually, all v does is replace an ordinary derivative () with the corresponding covariant one ().Conversely, given a connection map for a linear connection , there is a naturally induced covariant derivative on the smooth vector bundle , defined byThe scaling equivariance of v is critical for showing that this map actually defines a covariant derivative. Full type safety should be observed here; by the contravariance of the pullback of bundles (see (8)), , soand therefore as desired. This connection map construction of a covariant derivative gives (10) as an immediate consequence via the chain rule for the tangent map. The following construction is an abstraction of taking partial derivatives of a function, inspired by ([
11], p. 277). Instead of taking partial derivatives with respect to individual coordinates, partial covariant derivatives along distributions over the base manifold are formed, where the distributions (subbundles) decompose the base manifold’s tangent bundle into a direct sum. Such a construction conveniently captures the geometry of maps with respect to the geometry of its domain.
Proposition 16 (Partial covariant derivatives)
. Let , and for each let be a smooth vector bundle. If, for each , such that is a smooth vector bundle isomorphism over , then there exist unique sections for each such thatThis decomposition of provides what will be called partial covariant derivatives of L (with respect to the given decomposition). Proof. The following equivalences provide a formula for directly defining
.
Existence and uniqueness is therefore proven. □
Corollary 3 (Horizontal/vertical derivatives). Let as before. If is a connection map, and if is smooth, then there exist unique and such that .
It should be noted that the basepoint-preserving issue discussed in
Section 3.7 plays a role in choosing to use the tensor field formulation of
and
. In particular, without preserving the basepoint (via the
-pullback of
and
E to form
and
), the map
would not be a smooth bundle isomorphism, and the horizontal and vertical derivatives would be maps of the form
and
, but that, critically, are
not sections of smooth vector bundles, and can only claim to be smooth [fiber] bundle morphisms. Derivative trivializations will be central in calculating the first and second variations of an energy functional having Lagrangian
L (see (1) and (2)).
3.10. Curvature and Commutation of Derivatives
A ubiquitous consideration in mathematics is to determine when two operations commute. In the setting of tensor calculus, this often manifests itself in determining the commutativity (or lack thereof) of two covariant derivatives. Here, “covariant derivatives” may refer to both linear covariant derivatives and the tangent map operator (see (3)). This unified categorization of derivatives will now be leveraged to show that certain fiber bundles are flat (in a sense analogous to the vanishing of a curvature endomorphism) with respect to particular covariant derivatives. This reduces the work often done showing commutativity of derivatives in the derivation of the first variation of a function in the calculus of variations to the simple statement that a particular tensor field is symmetric, which is comes as a corollary to the aforementioned flatness.
In this section, the symbol ∇ may denote
or
, depending on context. This eases the expression of repeated covariant derivatives, such as the covariant Hessian of a section (see below), and is an example of telescoping notation as discussed in
Section 3.3.
If
defines a smooth [fiber] bundle whose space of sections
has two repeated covariant derivatives defined, if
, and if
is a symmetric linear covariant derivative (meaning
for
), then the tensor contraction
is an expression measuring the non-commutativity of the
X and
Y derivatives of
. The quantity
will be called the
covariant Hessian of
, because it generalizes the Hessian of elementary calculus; it contains only second-derivative information, and in the special case seen below, it is symmetric in the argument components. It should be noted that if
is the vector bundle such that
, then
. Intentionally leaving the ∇ and · symbols undecorated in preference of contextual interpretation, unwinding the expression above gives
which is syntactically identical to the common definition for the [Riemannian] curvature endomorphism
. In the traditional setting, where
is a linear covariant derivative on vector bundle
E, the curvature endomorphism takes the form of a tensor field
. In this setting however, because
may be nonlinear (for example,
when
M and
S are manifolds), such a tensorial formulation does not generally exist. Instead,
defines a second-order covariant differential operator (“covariant” meaning tensorial in the
X and
Y components). Put differently,
which will be called the (possibly nonlinear)
curvature operator, which in particular measures the non-commutativity of the
X and
Y derivatives of
. If
is identically zero, then the bundle
E is said to be
flat with respect to the relevant connections/covariant derivatives.
There are two particularly important instances of flat bundles. The first is the trivial line bundle defined by
(whose space of smooth sections, as discussed in
Section 3.4, is naturally identified with
;
S is a smooth manifold). In this case,
and
is the object referred to in most literature as the covariant Hessian of
f. Here,
is a real-valued function on
S.
Proposition 17 (Symmetry of covariant Hessian on functions). Let S be a smooth manifold and let be a symmetric covariant derivative. If , then is a symmetric tensor field (i.e., it has a symmetry). Here, the covariant derivative on is as defined above.
Proof. Let
. Recall that
. Then
Because
is pointwise-arbitrary in
, this shows that
is symmetric. Equivalently stated,
is identically zero, and therefore the relevant bundle is flat. □
The second important case involves the nonlinear covariant derivative
on
. Here, if
, then
so
.
Proposition 18 (Symmetry of covariant Hessian on maps). Let M and S be smooth manifolds and let and be symmetric covariant derivatives. If , then is a tensor field which is symmetric in the two components (i.e., it has a symmetry). Here, the covariant derivative on is as defined above.
Proof. Let
and
, so that
. Then
By definition,
, which cancels out the other term. By (17),
is symmetric, so the final term is zero. Because
is pointwise-arbitrary in
and
X and
Y are pointwise-arbitrary in
, this shows that
is identically zero, so the bundle defined by
, whose space of sections is identified with
, is flat, and therefore
is symmetric in its two
components. □
The construction used in (16) can be applied to nonlinear as well as linear covariant derivatives to considerable advantage. For example, if
, where
are smooth manifolds and
and
, then define
and
by
This gives a convenient way to express partial covariant derivatives, which will be used heavily in
Section 4 in calculating the first and second variations of an energy functional. Note that in this parlance,
is the full tangent map
.
Defining second partial covariant derivatives
,
,
and
by
the symmetry of the covariant Hessian of
can be used to show the various symmetries of these second derivatives.
Proposition 19 (Symmetries of partial covariant derivatives)
. With ψ and its second partial covariant derivatives as above,and (having analogous type) are -symmetric (i.e., and ) and the mixed, second partial covariant derivativesare mutually -symmetric (i.e., ). Proof. Let
. If
and
, then
Because
and
are pointwise-arbitrary in
and
respectively, this implies that
.
Analogous calculations (setting and and then separately setting and ) show that and . □
There are two final results regarding the second covariant derivative that will be especially useful in the calculation of the first and second variations of an energy functional (see (1) and (3)).
Proposition 20 (Chain rule for covariant Hessian)
. Let define a bundle having a first and second covariant derivative (i.e., a section of E can be covariantly differentiated twice). If and , then Proof. Let
. Then
Because
X is pointwise-arbitrary in
, this establishes the desired equality. □
Proposition 21 (Pullback curvature endomorphism)
. Let define a vector bundle having first and second covariant derivatives. If , then Proof. Note that
. Let
and let
, so that
. Then
and because
and
are pointwise-arbitrary in their respective spaces, this establishes the desired equality. □
A common operation is to evaluate a covariant derivative along a single tangent vector. One can express a single tangent vector as a section of a particular pullback bundle, the map being the constant map evaluating to the basepoint of the vector. This allows the richly typed formalism of pullback bundles to be used to evaluate derivatives at a point, particularly noting that this safely deals with the overloading of the natural pairing operator · (see
Section 3.5).
Proposition 22 (Evaluation commutes with non-involved derivatives)
. Let A and B be smooth manifolds and let for some smooth bundle having a covariant derivative . If and the map represents evaluation at b, theni.e., evaluation in B commutes with a derivative along A. Proof. Let
, and let
and
. Then
and because
X is pointwise-arbitrary in
, this implies that
as desired. □
Proposition 23. Let be smooth manifolds, let be smooth, let and , and let . If and , then Proof. The conditions
and
imply that
in the product covariant derivative. Then since
, it follows that
where
denotes the identity tensor field on
, and therefore
For the main calculation,
as desired. □