1. Introduction
A Riemannian manifold
is conformally flat if every point has a neighborhood which is conformal to an open set in the Euclidean space
. A hypersurface in the Euclidean space
is said to be conformally flat if it is with respect to the induced metric. The dimension of the hypersurface seems to play an important role in the study of conformally flat hypersurfaces. For
, the immersed hypersurface
is conformally flat if and only if at least
of the principal curvatures coincide at each point according to the result of Cartan-Schouten [
1,
2]. Cartan-Schouten’s result is no longer true for three-dimensional hypersurfaces. In fact, the number of distinct principal curvatures of a three-dimensional conformally flat hypersurface may be
, or 3. Lancaster [
3] gave some examples of conformally flat hypersurfaces in
having three different principal curvatures. These standard examples come from cones, cylinders, or rotational hypersurfaces over a surface with a constant Gaussian curvature in three-sphere
, Euclidean three-space
, or hyperbolic three-space
, respectively. For
, the existence of isothermal coordinates means that any Riemannian surface is conformally flat.
In [
1], E. Cartan gave a local classification for conformally flat hypersurfaces for dimensions
, and Cartan proved that if
, then
is a conformally flat hypersurface if and only if
f is a branched channel hypersurface (see [
1], Section 24). In [
4], Do Carmo, Dajczer, and Mercuri studied the diffeomorphism types of the compact conformally flat hypersurfaces of dimensions
. Pinkall in [
5] studied the intrinsic conformal geometry of compact conformally flat hypersurfaces. Suyama in [
6] explicitly constructed compact conformally flat hypersurfaces in space forms using co-dimension single foliation with
spheres.
If a conformally flat hypersurface
in
has two distinct principal curvatures, then it shares the same geometric properties with the conformally flat hypersurfaces of dimensions
. For a generic conformally flat hypersurface
, according to the theorem in [
1], there exists an orthogonal curvature line coordinate system at each point of such hypersurfaces. In a series of papers [
7,
8,
9,
10], Jeromin and Suyama studied Guichard’s nets on generic conformally flat hypersurfaces. In other papers [
6,
11,
12], Suyama locally classified the generic conformally flat hypersurfaces in terms of the first fundamental form. The (local) classification of these hypersurfaces is far from complete. For generic conformally flat hypersurfaces, whether all principal curvatures are distinct is an open condition, and thus the geometry of generic conformally flat hypersurfaces is more fruitful.
It is known that the conformal transformation group of
is isomorphic to its Möbius transformation group if
. As conformal invariant objects, the Möbius geometry is the most suitable framework for the study of conformally flat hypersurfaces. The Möbius form
C is an important invariant in the Möbius geometry of hypersurfaces, which determines whether the Möbius second fundamental form is a Codazzi tensor. For a hypersurface
, let
be an orthonormal basis with respect to the induced metric
with the dual basis
, and let
, and
be the second fundamental form and the mean curvature of
f, respectively. Let
. Then, the Möbius form
C is defined as follows:
In [
13], Li Haizhong and Wang Changping classified the surfaces with vanishing-Möbius forms (alternatively, see [
14]). Guo Zhen and Li Fengjiang studied surfaces with closed Möbius forms in [
15]. For high-dimensional hypersurfaces, Lin Limiao and Guo zhen classified the hypersurfaces with two or three constant Möbius principal curvatures and a closed Möbius form in [
16,
17]. In the study of other special hypersurfaces, the Möbius form vanishing is a necessary condition (see [
17,
18,
19]). In [
16], Lin and Guo classified the conformally flat hypersurface of dimensions
with a closed Möbius form in Möbius geometry. Lin and Guo’s results are still valid for three-dimensional conformally flat hypersurfaces in
if the number of distinct principal curvatures of the hypersurfaces is two. In this paper, using the framework of Möbius geometry, we investigate the generic conformally flat hypersurfaces in
. First, we classify the generic conformally flat hypersurfaces with a vanishing Möbius form:
Theorem 1. Let be a generic conformally flat hypersurface. If the Möbius form vanishes, then f is locally Möbius equivalent to a cone over a homogeneous torus in three-sphere .
Let
be an immersed hypersurface with the three distinct principal curvatures
, and
. Then, the Möbius curvature
is only an extrinsic Möbius invariant. If the Möbius curvature of a generic conformally flat hypersurface is constant, then we prove that the Möbius form vanishes, and thus we have the following results:
Proposition 1. Let be a generic conformally flat hypersurface. If the Möbius curvature is constant, then f is locally Möbius equivalent to a cone over a homogeneous torus in three-sphere .
Second, we investigate the global behavior of compact generic conformally flat hypersurfaces with Möbius invariants. The Ricci curvature of the hypersurface is called nonpositive pointwise if the Ricci curvature for all . Similarly, a nonnegative pointwise Ricci curvature means that for all :
Theorem 2. Let be a generic conformally flat hypersurface. If the hypersurface is compact, then the Ricci curvature of the Möbius metric cannot be nonpositive pointwise or nonnegative pointwise.
Theorem 3. Let be a generic conformally flat hypersurface and g and A be the Möbius metric and the Blaschke tensor, respectively. Let . If the hypersurface is compact, thenwhere and R denote the norm of the Ricci curvature and the scalar curvature of the Möbius metric g, respectively. Corollary 1. Let be a generic conformally flat hypersurface and g be its Möbius metric. If the hypersurface is compact, then Corollary 2. Let be a generic conformally flat hypersurface and g be its Möbius metric. If the hypersurface is compact, then Remark 1. There exist many examples of generic conformally flat hypersurfaces in (see [6,7,8,11] or Section 3), but these examples are local and noncompact. It is difficult to build a compact, generic conformally flat hypersurface with these local examples. Theorems 2 and 3 show that there are many obstructions to constructing a compact, generic conformally flat hypersurface in , and it is relatively rare. Remark 2. Due to conformal invariant objects, the theory of conformally flat hypersurfaces is essentially the same whether it is considered in the space forms , , or . In fact, there exists conformal diffeomorphism between the space forms. The -dimensional hyperbolic space defined by The conformal diffeomorphisms σ and τ are defined bywhere is the hemisphere in , in which the first coordinate is positive. Under conformal diffeomorphisms σ and τ, the conformally flat hypersurfaces in the space forms are equivalent to each other. This paper is organized as follows. In
Section 2, we review the elementary facts about the Möbius geometry of hypersurfaces in
. In
Section 3, we investigate the Möbius invariants of generic conformally flat hypersurfaces in
and prove Theorem 1 and Proposition 1. In
Section 4, we investigate the global behavior of compact generic conformally flat hypersurfaces in
and prove Theorems 2 and 3.
2. Möbius Invariants of Hypersurfaces in
In this section, we define Möbius invariants and give a congruent theorem of hypersurfaces in
. For details, we refer the reader to [
20,
21].
Let
denote the
-dimensional Euclidean space and a dot · represent its inner product. The
-dimensional sphere is
. The hypersphere
in
with acenter
and radius
is given by
Let
. By taking
such that
, a line
l that passes through the point
o intersects the sphere
in two points
. Now we define the Möbius inversion
for the point
as follows:
Clearly, . When the point o is at infinity, the Möbius inversion is indeed a reflection and thus an isometric transformation of :
Proposition 2 ([
22])
. The Möbius transformation group is generated by Möbius inversions . Let
be the Lorentz space (i.e.,
), with the scalar product
defined by
for
.
Let
be the set of an invertible
matrix. Then, the Lorentz orthogonal group
is defined by
where
denotes the transpose of the matrix
T,
and
I is the
unit matrix.
The positive light cone is
and
is the subgroup of
defined by
Proposition 3 ([
23])
. Let , where Q is an matrix. Then, if and only if It is well known that the subgroup
is isomorphic to the Möbius transformation group
. In fact, for any
we can define the Möbius transformation
as
Then the map is a group isomorphism.
Let
be an isometric transformation of
. Then,
, and
Thus, is a subgroup.
The
-dimensional sphere
is diffeomorphic to the projective light cone
such that
The diffeomorphism
is given by
The group
acts on
as follows:
With conformal map
we can obtain the conformal invariants of hypersurfaces in
. Let
be a hypersurface without umbilical points and
be an orthonormal basis with respect to the induced metric
with the dual basis
. Let
and
be the second fundamental form and the mean curvature of
f, respectively. To study the Möbius geometry of
f, as in [
20,
21], one considers the Möbius position vector
of
f as follows:
where · denotes the Euclidean inner product in
.
Theorem 4 ([
21])
. Two hypersurfaces are Möbius equivalent if and only if there exists T in the Lorentz group such that It follows immediately from Theorem 4 that
is a Möbius invariant called the Möbius metric of
f.
Let
be the Laplacian with respect to
g. We define
which satisfies
Let
be a local orthonormal basis for
with a dual basis
. We write
, and then we have
Let
be the mean curvature sphere of
f, written as
where
is the unit normal vector field of
f in
. Thus,
forms a moving frame in
along
. We will use the following range of indices in this section:
. We can write the structure equations as follows:
where
is the connection form of the Möbius metric
g and
. The tensors
are called the Blaschke tensor, the Möbius second fundamental form, and the Möbius form of
f, respectively. The covariant derivatives of
are defined by
The integrability conditions for the structure equations are given by
Here, denotes the curvature tensor of g, and is the Möbius scalar curvature. We know that all coefficients in the structure equations are determined by when . Thus, we have the following:
Theorem 5 ([
21])
. Two hypersurfaces and are Möbius-equivalent if and only if there exists a diffeomorphism which preserves the Möbius metric and the Möbius second fundamental form. We would also like to recall from [
20,
21] how
can be calculated in terms of the geometry of
f in
:
where the Hessian and ∇ are with respect to
. The eigenvalues of
B are called the Möbius principal curvatures of
f. Let
be the Möbius principal curvatures and
be the principal curvatures of
f. Then, from Equation (
7), we have
Clearly, the number of distinct Möbius principal curvatures is the same as that of the principal curvatures of
f and
which confirms that the Möbius curvatures are Möbius invariants.
3. Some Local Properties of the Generic Conformally Flat Hypersurfaces
In this section, we recall some facts about the generic conformally flat hypersurfaces and prove Theorem 1 and Proposition 1.
Let
be a Riemannian manifold and ∇ be its Riemannian connection. The Riemannian curvature
tensor is defined as follows:
If the Riemannian curvature tensor vanishes (i.e., ), then the Riemannian metric g is flat.
Let
be a local orthonormal basis for
with a dual basis
. The Ricci curvature
and scalar curvature
R are defined by
The Schouten tensor
S is the symmetric
tensor, defined by
Let
, and
. Then, we have
The Weyl tensor
W is the
tensor defined by
With the Schouten tensor, the Weyl tensor also can be defined by
When the dimension , the Weyl tensor . The Weyl tensor is an algebraic curvature which is traceless.
Let
be the coefficients of the covariant derivative of
S. Then, the Cotten tensor
T is defined by
Definition 1. Two Riemannian metrics g and on the manifold are conformal if there is a smooth function ρ on such that
Let
be the Weyl tensor of the metric
. Then, we have
When the dimension
, let
be the Cotten tensor of the metric
. Then, we have
Definition 2. The Riemannian manifold is said to be conformally flat if, for any , there is an open set containing p and a smooth function such that the Riemannian metric is flat.
The following results are due to Weyl’s work:
Theorem 6. (i) Any two-dimensional Riemannian manifold is conformally flat. (ii) A three-dimensional Riemannian manifold is conformally flat if and only if the Cotten tensor . (iii) If the dimensions , then the Riemannian manifold is conformally flat if and only if the Weyl tensor .
Let
be a generic hypersurface. Since
f is without an umbilical point, the Möbius metric
g, the Möbius second fundamental form
B, the Blaschke tensor
A, and the Möbius form
C can be defined. We chose a local orthonormal basis
with respect to the Möbius metric
g such that
Let
be the dual basis of
. The conformal fundamental forms of
f are defined by
Using Equations (5) and (6), the Schouten tensor of
f with respect to the Möbius metric
g is
If the hypersurface
f is conformally flat, then
. By combining Equations (
1) and (
10), we obtain the following equation:
Using Equation (3), we have the following equations:
Using
and Equation (
10), we obtain
The following lemma is trivial under Equations (
13) and (
14) (alternatively, see [
8,
12]):
Lemma 1 ([
8,
12])
. Let be a generic hypersurface. Then, the following are equivalent:(1) The hypersurface is conformally flat;
(2) The Schouten tensor is a Codazzi tensor;
(3) The conformal fundamental forms , and are closed.
Next, we give the standard examples of generic conformally flat hypersurfaces in :
Example 1. Let be an immersed surface. We define the cylinder over u in aswhere is the identity map. If the surface u is of a constant Gaussian curvature, then the cylinder over u is conformally flat. Example 2. Let be an immersed surface. We define the cone over u in as If the surface u is of a constant Gaussian curvature, then the cone over u is conformally flat.
Let
be the circle with a radius
r and
be the circle with a radius
. Then, the homogeneous torus
is defined as follows:
The homogeneous torus x is a flat surface in , and thus the cone over the homogeneous torus is conformally flat.
Example 3. Let be the upper half-space endowed with the standard hyperbolic metric Let be an immersed surface. We define the rotational hypersurface over u in aswhere is the unit circle. If the surface u is of a constant Gaussian curvature, then the rotational hypersurface over u is conformally flat. Before the proof of Theorem 1, we need the following results, given in [
24]:
Theorem 7 ([
24])
. Let be an immersed hypersurface with three distinct principal curvatures. If f is of a constant Möbius sectional curvature c, then f is Möbius equivalent to a cone over a flat torus with a Möbius sectional curvature . Proof. Next, we prove Theorem 1. Since the Möbius form vanishes,
. Then, under Equation (
13), we have
Since
, then
Thus, we obtain
According to Equation (
14), we have
, which implies that the hypersurface is flat. Using Theorem 7, we finish the proof of Theorem 1. □
Proof. Next, we prove Proposition 1. Let
be a generic conformally flat hypersurface with the Möbius curvature
being constant. Let
be the Möbius principal curvatures and
Thus, combined with Equation (6), we have
which implies that the Möbius principal curvatures
are constant. From Equation (
13), we find that
, and with Theorem 1, we finish the proof of Proposition 1. □
4. Some Global Behavior of the Compact, Generic Conformally Flat Hypersurfaces
In this section, we investigate the global behavior of the compact, generic conformally flat hypersurfaces and give the proof of Theorems 2 and 3 as well as Corollaries 2 and 1.
Let be a generic conformally flat hypersurface. Let , and denote the Blaschke tensor, the Möbius second fundamental form, and the Möbius form, respectively. We say that the pair is admissible if the following are true:
- (1)
U is an open subset of ;
- (2)
is an orthonormal co-frame field on U with respect to the Möbius metric g;
- (3)
;
- (4)
.
Denote with
the dual frame field of
. Then, it is easily seen that
is admissible if and only if
is a unit principal vector associated with
for each
, and
is an oriented basis associated with the orientation of
. Denote with
the connection form with respect to
. Thus, under the admissible frame field
, we have
Now, we introduce two two-forms
and
on
. For every admissible co-frame field
, we set
If
and
are both admissible co-frame fields with
, then on
,
for every
, where
or
and
. Thus, we have
Therefore, the two forms and are well-defined on .
By combining
,
, and Equations (
13) and (
14), we obtain
Similarly, we can compute
and
. Therefore, we have
By using
and the same computation as
, we can obtain
where
. By combining Equation (
15) and Equation (
16), we have
If
is compact, then Equation (
17) implies that
Let
, and
be the Ricci curvatures. Then, we have
By combining
and
, we can derive that
Thus from Equation (
18), we have
From
and
, we find
, and thus we have
Proof. Now, we will prove Theorem 2. If the Ricci curvature of the Möbius metric is nonpositive pointwise ( or nonnegative pointwise), then
, and
(or
, and
). Thus, we have
From Equation (
19), we find that
, and thus the Möbius sectional curvatures
, and
are zero. Therefore, the hypersurface is noncompact under Theorem 7, which is a contradiction with the statement that the hypersurface is compact. Theorem 2 is proven. □
Proof. From
and
, we have
, and thus
If the equality holds pointwise on the global hypersurface
f in the above inequality, then
, and
on
f, which implies that the Ricci curvature has a sign. Under Theorem 2, we know this is impossible. Through combination with Equation (
19), Corollary 1 is proved. □
Proof. Using Equations (4) and (6), we have
On the other hand, Equation (5) implies that
where
denotes the norm of the Ricci curvature. By combining Equations (
21) and (
22), we can derive that
Let
denote the trace-free Blaschke tensor. Then,
. Thus, from Equation (
23), we have
Now, if the hypersurface
is compact, then
Therefore, we finish the proof of Theorem 3. □
Proof. Since
, we have
on
. If
, then
, and the hypersurface is an Einstein type. Since the dimension of the hypersurface is three, the sectional curvatures are constant. From the results in [
24,
25], we know that the hypersurface is locally Möbius equivalent to a cone over a homogeneous torus
in three-sphere
, which is a contradiction with the idea that the hypersurface is compact. Thus,
and
Since
under Equation (6), by combining this with Equation (
25), we have
Thus, Corollary 2 is proven. □