1. Introduction
Riemannian maps, first presented by A.E. Fischer [
1] in 1992, are specific types of mappings between Riemannian manifolds that preserve certain geometric structures or properties. These maps play a crucial role in differential geometry by facilitating the comparison and analysis of geometric features across different manifolds. These maps extend the familiar concepts of Riemannian submersions and isometric immersions. Riemannian maps are used in various mathematical and applied contexts to understand the relationships between different geometric spaces, analyze geometric structures, and solve problems in physics, computer science, and beyond, where geometric structures play a key role.
Let
and
be two Riemannian manifolds with
and
. Suppose there is a smooth mapping
, where
. At any point
, we define the kernel space of
by
and the orthogonal complementary space to
by
(horizontal space). Consequently, the tangent space
at
x of
B can be expressed as the direct sum of these two spaces:
Consider the range of
, denoted by
, at point
x of
B and the orthogonal complementary space to
by
in the tangent space of
F at
x, denoted by
. Since
, we have
. Hence,
can be decomposed as follows:
Definition 1. A smooth map with is said to be a Riemannian map at if the horizontal restrictionis a linear isometry, that is,satisfies the equationfor any vector fieldsandtangent to H. Furthermore, if P acts as a Riemannian map at each point, then P is known as a Riemannian map (shortly, RM). Remark 1. It easy to observe that RMs are isometric immersions if and that they are Riemannian submersions if .
Remark 2. A notable property of RMs was shown by Fischer [1], who stated that they satisfy the generalized eikonal equation, which is a bridge between geometric optics and physical optics. In summary, Riemannian maps are versatile tools that facilitate the transfer and analysis of geometric, topological, and physical properties across different spaces, finding applications in a wide range of scientific and engineering disciplines: RMs can be used to understand the deformation and mapping of spacetime in the context of general relativity; they are also used to study shapes and their transformations in computer graphics and computer vision and help us to understand the relationships between different geometric structures, as well as having many more uses.
On the other hand, the renowned embedding theorem devised by Nash provided significant inspiration for defining precise relationships between the intrinsic and extrinsic invariants of Riemannian submanifolds. The derivation of optimal inequalities for submanifolds in diverse ambient spaces is primarily influenced by isometric immersions. The Wintgen inequality is a fundamental result in the differential geometry of submanifolds, particularly in the study of surfaces in higher-dimensional spaces. It outlines the relationship between the intrinsic and extrinsic geometry of a submanifold. In [
2], P. Wintgen demonstrated that for any surface in
, the Gauss curvature
K, the squared mean curvature
, and the normal curvature
always satisfy the following inequality:
The equality holds if and only if the ellipse of the curvature of a surface in
is a circle.
Geometric Interpretation: The Wintgen inequality expresses a balance between the intrinsic curvature (the Gaussian curvature) and the extrinsic curvature (the mean curvature and the traceless part of the second fundamental form) of the surface. It essentially states that the intrinsic curvature cannot be too large compared to the extrinsic curvature measures.
Example 1. Consider a minimal surface in , where the mean curvature . The Wintgen inequality simplifies to the following:In this case, the Gaussian curvature K is non-positive, reflecting the fact that minimal surfaces tend to have saddle points. P.J. De Smet, F. Dillen, L. Verstraelen, and L. Vrancken [
3] proposed the DDVV conjecture, often known as the generalized Wintgen inequality, the DDVV inequality, or the normal scalar curvature conjecture, for a submanifold
of a real space form
of constant sectional curvature
c. The conjecture is expressed as follows:
where
represents the (intrinsic) normalized scalar,
denotes the (extrinsic) normalized normal scalar, and
represents the (extrinsic) squared mean curvatures of
M. It is worth mentioning that number of geometers achieved partial results concerning the DDVV conjecture (for example, [
4]). However, the conjecture, in its general form, was independently proven in [
5] and in [
6].
The DDVV inequality, initially applied to submanifolds in real space forms, has seen significant extensions and applications across a variety of geometric contexts thanks to the efforts of many researchers: It was notably extended by I. Mihai for Lagrangian and slant submanifolds in complex space forms [
7], alongside extensions to Legendrian submanifolds in Sasakian space forms [
8]. M.E. Aydin, A. Mihai, and I. Mihai [
9] further broadened its application to statistical submanifolds in statistical manifolds of constant curvature. J. Roth [
10] developed a version of the DDVV inequality for submanifolds of warped products in the form
, where
I is an interval and a smooth function
that never vanishes. C. Murathan and B. Sahin [
11] sourced inspiration from Roth and proved a general Wintgen inequality for the statistical submanifolds of statistical warped product manifolds. The generalized Wintgen inequality for a Legendrian submanifold in standard warped product manifolds, specifically in
-Kenmotsu manifolds, was derived by A.N. Siddiqui, et al. [
12]. In advancing this topic, the generalized Wintgen inequality is established for statistical submanifolds in Hessian manifolds of constant Hessian curvature [
13]. Indeed, a number of geometers have successfully derived the Wintgen inequality in different settings, and these works have contributed to the deepening of our understanding of curvature relations and geometric inequalities in these specialized contexts.
Therefore, the main aim of this article is to extend this conjecture for an RM with a real space form as a target manifold. Also, we discuss the same for Riemannian manifold of quasi-constant curvature (and of nearly quasi-constant curvature) as a target manifold. Moreover, we extend this inequality to the case of Riemannian maps in which the target manifold is a complex space form.
2. Preliminaries
For a Riemannian map (shortly, RM)
its differential, denoted
acts as a section of the bundle
on
B. Here,
represents the pullback bundle, with fiber
for each point
. The bundle
has an induced connection
D from the Levi-Civita connection
on
B. Then, the second fundamental form of
P can be expressed as [
14]
for any
, where
refers to the pullback connection. This second fundamental form is known to be symmetric. Additionally, it has been established that
has no components in
, provided that
, where
denotes the subbundle of
with fiber
,
.
The Levi-Civita connection on
F is denoted by
and its pullback along
P. The Weingarten formula for
P is given by [
14]
where
is the orthogonal projection of
,
and
, and
is the shape operator defined on
. The second fundamental form is bilinear, and through direct calculation, it is shown that [
14]
for all
and
. Given that
is symmetric, this implies that
is a symmetric linear transformation of
.
Consider
and
as the curvature tensor fields of
and
, respectively. Consequently, the Gauss equation can be expressed as follows [
14]:
for
.
The Ricci equation is expressed as [
14]
for
. Here,
.
For any
, consider
to be an orthonormal basis for
, where
forms an orthonormal basis for
and
forms an orthonormal basis for
. The scalar curvature, denoted as
, on
is calculated by using
and the normalized scalar curvature of
is defined by
A new notion of normal scalar curvature, denoted as
, is defined for every point
as follows:
where
represents an orthonormal basis of
.
3. Riemannian Maps to Real Space Forms
Consider the target manifold as a real space form, which is an
s-dimensional Riemannian
(shortly, Rie.) manifold
with constant sectional curvature
c, denoted by
. In this context,
of
is expressed as
for the vector fields
on
F.
We prove the following result by utilizing Lemma 2.1 from [
3], which states that
Lemma 1. Let for . Thenand equality holds if and only ifIfthenfor integers k and l such that and . Theorem 1. Let P be an RM from an r-dimensional Rie. manifold onto a -dimensional real space form with . Then, we have Proof. We choose an orthonormal basis of
of
and choose
orthogonal to
, which is in the direction of the mean curvature vector. Then, we compute (
15) by inputting
,
,
,
as follows:
Thus, (
11) reduces to
where we use (
9) and the notation
. So
Then, we can compute the following:
Therefore, by employing (
10) and taking into account our selected basis, we conclude that
To introduce inequality into the above formula, we derive
Lemma 1 and the relation
yield
Applying (
11) and (
14) to the above inequality, we arrive at
from which we deduce (2). □
In light of the aforementioned main theorem, we can say the following:
Theorem 2. Let P be an RM from an r-dimensional Rie. manifold onto an s-dimensional real space form with . Then, we have Proof. We choose orthonormal bases of
and
of
and
, respectively. Then, by the Gauss equation, we have
Next, we write
as
and use the following inequality from [
5]:
Then, from (
14), (
19), and inequality (
20), and hence together with (
18), we finally arrive at
which further gives
□
To support Theorem 2 and (1), let us revisit several examples, as outlined below:
Example 2. We recall Example 1 from [15]. Consider the standard Rie. submersion between the 15-dimensional sphere of constant sectional curvature 1 and the 8-dimensional sphere of constant sectional curvature 4 and radius , with totally geodesic fibers and the totally umbilical isometric immersion . Then, we have an RM that satisfies Theorem 2. Example 3. Consider Hopf fibration and the canonical embedding P of a hypersurface of to be . Then, we have a totally umbilical RM that satisfies Theorem 2 (see [16]). Corollary 1. Let P be a RM from an onto an s-dimensional Euclidean space with . Then, we have By applying Theorem 2 in the specific case in which
P acts as an isometric immersion, we are able to retrieve the traditional DDVV inequality for submanifolds in real space forms [
3]. Since
P is an isometric immersion, we have
, and hence, we deduce the following result using Theorem 2:
Corollary 2. Let P be an isometric immersion from a into . Then, we have 6. Concluding Remarks
Riemannian maps are used to study the relationships between different geometric structures on manifolds. They help us with understanding how geometric properties (like curvature) are transferred from one manifold to another and have applications in theoretical physics, particularly in general relativity and the study of space-time manifolds.
The field of Riemannian maps represents a recent area of research, extending beyond traditional studies of Riemannian submanifolds and Riemannian submersions. While much attention has been given to curvature relationships concerning submanifolds and Riemannian submersions, this paper aims to explore Riemannian maps through the lens of curvature relations. The examples that we have presented in this article illustrate the efficacy of our results. By demonstrating the consistency of our findings in specific examples, we offer compelling evidence for the broad applicability and resilience of our results. Although we have established Wintgen inequality for Riemannian maps, there are still many research problems to investigate for interested readers. In this direction, one can extend the derived inequality in the case of Riemannian maps to known space forms, for example, Sasakian space forms and quaternionic space forms (see [
22,
23]). Furthermore, we want to intersect other techniques and theories to obtain more new results. For example, the new works relevant to singularity theory [
24], soliton theory [
25,
26], hypersurfaces problems [
27,
28], Chen–Ricci inequality [
29,
30], submanifolds theory [
31,
32], tangent bundle problems [
33,
34], etc. are useful for future research. We will combine the techniques and results mentioned above to explore new approaches for obtaining new results in future research.