1. Introduction
The study of Finslerian hypersurfaces and their classification represents a significant advancement in the field of Finsler geometry, a branch of differential geometry that generalizes Riemannian geometry by allowing the metric to depend not only on position but also on direction. The notion of Finslerian hypersurfaces was first introduced by the eminent mathematician Matsumoto, who provided a systematic classification of these hypersurfaces into three distinct types: hyperplanes of the first kind, second kind, and third kind. This classification was based on the geometric and algebraic properties of the hypersurfaces and their relationship to the underlying Finsler metric. Matsumoto’s work laid the groundwork for subsequent research, inspiring numerous mathematicians to explore the properties of these hypersurfaces under various modifications and generalizations of the Finsler metric. These investigations, as documented in references [
1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12], have uncovered a rich array of geometric properties, deepening our understanding of the intrinsic structure of Finslerian hypersurfaces and their applications in geometry and physics.
In addition to his contributions to the theory of Finslerian hypersurfaces, Matsumoto also introduced the concept of an
-metric [
13], which has since become a central topic of research in Finsler geometry. An
-metric is a generalization of the classical Riemannian metric, where
represents the Riemannian part of the metric and
is a 1-form that introduces a directional dependence. The exponential metric, expressed as
, is a unique form of the
-metric that has attracted significant attention due to its elegant structure and its connections to theoretical physics and cosmology. The exponential metric has been extensively examined by various authors [
14,
15,
16,
17,
18,
19,
20], who have explored its geometric properties under different conditions. One notable feature of the exponential metric is its relationship to Rander’s metric. Specifically, under certain transformations, the exponential metric can be reduced to Rander’s metric, which has significant applications in theoretical physics, particularly in the study of spacetime geometries and cosmological models. This connection highlights the importance of
-metrics not only in pure mathematics but also in applied fields.
A hypersurface is a generalization of the concept of hyperplane. It is defined as follows.
Definition 1. A sub-manifold of dimension is termed a hypersurface of an enveloping manifold of dimension n, and the co-dimension of the hypersurface is one.
If , then the space is termed a subspace of , and serves as an enveloping space for . Particularly, if , then is referred to as a hypersurface of the enveloping space .
Hence, a hypersurface
of the manifold
can be parametrically described by the equation
where
u represent the Gaussian coordinates on the hypersurface
.
If
represents the supporting line element at a point
on the hypersurface
, tangential to
, then we have
Thus,
is regarded as the supporting element of
at a point
. Considering the function
, which generates a Finsler metric on
, we obtain an
-dimensional Finsler space
.
In Finsler geometry, a hyperplane can be classified into different types based on its geometric properties. Below are some examples of such hypersurfaces:
Example 1. A sphere can be visualized as a three-dimensional structure and is an example of a two-dimensional manifold embedded in three-dimensional space. Thus, a hypersurface on this sphere would be a one-dimensional curve.
Example 2. A torus can be visualized as a three-dimensional structure and is an example of a two-dimensional manifold embedded in three-dimensional space, and a spiral curve would represent the hypersurface.
In
Figure 1, the first image shows the sphere as the three-dimensional space, with the curve (red lines) representing a hypersurface within it. The second image features a torus as the three-dimensional space, where the spiral curve (red line) represents a hypersurface within that structure.
Example 3 (Hyperplane of first kind). A hyperplane in a three-dimensional Finsler space is classified as a “hyperplane of the first kind" if it intersects with a given curve or surface within that space.
Example 4 (Hyperplane of second kind). A “hyperplane of the second kind" in Finsler geometry is a hyperplane (represented by a surface) that does not intersect with a given curve or surface in the three-dimensional Finsler space.
In
Figure 2, the first image shows a hyperplane intersecting with a curve, and this intersection is classified as a “hyperplane of the first kind”. In the second image, the surface is shifted upward by adding 5 to the z-values, preventing any intersection with the curve. This is referred to as the “hyperplane of the second kind”.
Example 5 (Hyperplane of third kind)
. We consider a flat plane embedded in 3D space. This plane has zero curvature, and its normal vector remains constant. This serves as a simplified analogy for a hyperplane of the third kind in Finsler geometry, where the conditions of vanishing curvature and tensor create a similar “flatness" within the Finsler space (Figure 3). Thus “hyperplane of the third kind” is essentially “flat” with respect to the ambient Finsler space.
In this paper, we introduce and analyze a novel -metric defined as . We refer to this metric as the Rander’s-type exponential -metric due to its structural similarity to Rander’s metric combined with an exponential factor. This metric represents a natural extension of the classical Rander’s metric and the exponential metric, combining their features in a way that opens up new avenues for geometric exploration. Our primary focus is on investigating the intrinsic properties of this Finsler space, particularly the conditions under which its hypersurfaces exhibit characteristics of hyperplanes of the first, second, and third kind. In Theorem 3, we derive the necessary and sufficient conditions for the hypersurfaces of the Rander’s-type exponential -metric to be classified as hyperplanes of the first kind. These conditions are expressed in terms of the geometric invariants of the Finsler space and provide a clear characterization of the hypersurfaces in this category. Similarly, in Theorem 4, we establish the conditions under which the hypersurfaces exhibit properties of hyperplanes of the second kind. These conditions involve a deeper analysis of the interplay between the Riemannian part and the 1-form in the metric. Finally, in Theorem 5, we address the case of hyperplanes of the third kind, identifying the specific geometric constraints that must be satisfied for the hypersurfaces to fall into this classification.
In addition to the classification of hypersurfaces, we also investigate the reducibility of the Cartan tensor for these hypersurfaces. The Cartan tensor is a fundamental object in Finsler geometry, encoding information about the anisotropy of the Finsler metric. Its reducibility, or the extent to which it can be decomposed into simpler components, provides valuable insights into the geometric structure of the Finsler space. In Propositions 1–3, we examine the reducibility of the Cartan tensor in various forms, focusing on the hypersurfaces associated with the Rander’s-type exponential -metric. These propositions reveal the conditions under which the Cartan tensor can be reduced to simpler forms, shedding light on the geometric behavior of the hypersurfaces and their relationship to the underlying metric.
Through this detailed analysis, we aim to contribute to the broader understanding of Finslerian hypersurfaces and their geometric properties, particularly in the context of the newly introduced Rander’s-type exponential -metric. Our results not only extend the existing theory of Finslerian hypersurfaces but also provide new tools for exploring the geometric and physical implications of -metrics. By uncovering the intrinsic properties of this metric and its hypersurfaces, we hope to inspire further research into the rich and diverse world of Finsler geometry.
2. Preliminaries
In this study, we investigate an
n-dimensional Finsler space represented as
. Here,
consists of an n-dimensional differentiable manifold
coupled with a fundamental function
that takes on a Rander’s-type exponential form within a unique Finsler space metric, expressed as
Taking partial derivatives of Equation (
1) with respect to
and
yields
where
.
In the Finsler space
, the normalized support element
and the angular metric tensor
are defined as, per reference [
21],
where
. For the fundamental metric Function (1) above, the constants are
The fundamental metric tensor
and its corresponding reciprocal tensor
for
can be found in reference [
21].
where
The reciprocal tensor
of
is given by
where
The hv-torsion tensor
is provided in reference [
10].
where
Here,
represents a non-zero covariant vector that is orthogonal to the support element
.
Given the components
representing the Christoffel symbols of the associated Riemannian space
, and using
to represent the covariant derivative with respect to
determined by these Christoffel symbols, we now introduce the following definition:
where
.
The Cartan connection of
, represented as
, defines the special Finsler space. The difference tensor
is given by
where
In this context, the symbol ‘
’ represents contraction with
, excluding the elements
,
, and
.
3. Cartan Connection for the Hypersurface of a Finsler Space
If
represents a hypersurface of
defined by
, with
, and if the supporting element
of
is tangent to
[
21], then
The metric tensor, the hv-tensor, a unit normal vector, the angular metric tensor, and the connection between projection factors and their inverses for a Finslerian hypersurface
[
21] at a point
are detailed as follows:
.
The Cartan connection
associated with the Finslerian hypersurface
is expressed as
where
and
Note: The tensorial quantities
and
are identified as the second fundamental v-tensor and the normal curvature vector, respectively.
Moreover, the second fundamental h-tensor
can be represented as, per [
21],
In this context,
Given the above expression, it is evident that the tensorial quantity
is non-symmetric, leading to
The covariant derivatives of the projection factor
with respect to the h- and v-directions of
can now be articulated as
When we contract
and
with
, the result is
Hence, the crucial findings for the Finslerian hypersurface [
21] that we will utilize in our current study are as follows.
Lemma 1. The normal curvature tensor becomes zero in all cases if and only if the normal curvature vector vanishes on a Finslerian hypersurface .
Lemma 2. In a scenario where symbolizes a Finsler space and signifies its hypersurface, the hypersurface is classified as a hyperplane of the first kind solely when the normal curvature vector completely disappears.
Lemma 3. Given a Finsler space denoted by and its corresponding hypersurface , the hypersurface is categorized as a hyperplane of the second kind only under the condition that both the normal curvature vector and the second fundamental h-tensor vanish completely.
Lemma 4. In the context where stands for a Finsler space and denotes its hypersurface, the hypersurface is classified as a hyperplane of the third kind if and only if the normal curvature vector, the second fundamental h-tensor, and the v-tensor vanish identically.
4. Hypersurface of a Finsler Space with Rander’s-Type Exponential Form of -Metric
In the context of a Finsler space featuring a Rander’s-type exponential
-metric expressed as
, where
denotes a Riemannian metric and the vector field
signifies the gradient of a scalar function
, we now explore a hypersurface
determined by the equation
, where
c stands for a constant [
10].
Obtained from the parametric representation
of
, we derive
The preceding demonstration illustrates that
represents the covariant components of a normal vector field of the hypersurface
. Moreover, we have
and the induced metric
of
is given by
which is a Riemannian metric.
Substituting
into Equations (2), (4) and (6) yields
From Equation (
5) we obtain
Therefore, traversing the Finslerian hypersurface
using Equations (20) and (17) results in
Thus, we have
where b is the length of the vector
.
Once more, by utilizing Equations (20) and (21), we obtain
Thus, we have the following theorem.
Theorem 1. The Riemannian metric induced on a Finsler hypersurface within a Finsler space , characterized by the Rander’s-type exponential metric described in Equation (1), is delineated in Equation (18). Moreover, the scalar function is defined by Equations (21) and (22). The angular metric tensor
and the fundamental metric tensor
of
can be expressed as
By combining Equations (17), (23) and (13), it can be deduced that, if
represents the angular metric tensor of the Riemannian
, then, along
,
.
Thus, along .
Deriving from Equation (
8), we obtain
Then, the hv-torsion tensor becomes
In the Rander’s-type exponential form of the
-metric of a Finsler hypersurface
, it follows from Equations (13), (14), (16), (17) and (24) that we obtain
Hence, based on Equation (
14), it can be concluded that
is symmetric, leading to the following theorem.
Theorem 2. The v-tensor, representing the second fundamental form for the hypersurface within a Finsler space characterized by the Rander’s-type exponential metric described in Equation (1), is given by Equation (25). Simultaneously, the h-tensor is identified as symmetric. Now, from Equation (
17), we have
. Then, we have
Consequently, by utilizing Equation (
16) and the expression
, we obtain
Since
, we obtain
Thus, deriving from Equation (
26), we have
Since
is symmetric, upon contracting Equation (
27) with
and applying Equation (
12), we obtain
Once more, contracting Equation (
28) with
and employing Equation (
12), we arrive at
From Lemmas 1 and 2, along with Equation (
29), it becomes evident that a Finslerian hypersurface
within a Finsler space featuring Rander’s-type exponential metric as given in Equation (
1) is a first-kind hyperplane if
.
Given that represents the covariant derivative concerning within the Finsler space defined over , whereas denotes the covariant derivative concerning the Riemannian connection , it follows that is independent of . Consequently, we are inclined to examine the disparity , where .
Given that
constitutes a gradient vector, we can deduce from Equation (
9) that
Leveraging the aforementioned fact and Equation (
10), the difference tensor
can be articulated as
where
Considering Equations (19) and (20), the connection in Equation (
11) transforms into
, owing to Equation (
31), resulting in
.
Contracting Equation (
30) with
now yields
Once more, contracting the previous equation with respect to
gives
Considering Equation (
17) in the context of
, we obtain
Contracting Equation (
32) with
, we have
Given Equations (21), (22) and (25), and
, we obtain
Therefore, with the relation
, Equations (32) and (33) yield
As a result, Equations (28) and (29) can be expressed as
Therefore, the condition
is equivalent to
. Utilizing the fact that
, the condition
can be restated as
for a certain
. Hence, we can express this as
Combining Equations (17) and (35), we obtain
Hence, from Equation (
34), we obtain
; again, from Equations (31) and (35), we obtain
, and
.
Now, employing Equations (20), (21), (22), (25) and (30), we arrive at
Therefore, Equation (
27) simplifies to
Therefore, the hypersurface
exhibits umbilic properties.
Theorem 3. The necessary and sufficient condition for the hypersurface in a Finsler space with a Rander’s-type exponential-form metric given by Equation (1) to be a hyperplane of the first kind is precisely defined in Equation (35). The following is evident from Equation (
35).
Corollary 1. The second fundamental h-tensor for a Finsler hypersurface of a Finsler space equipped with a Rander’s-type exponential-form metric defined in Equation (1) is directly related to its angular metric tensor. According to Lemma 3, the hypersurface
qualifies as a hyperplane of the second kind when and only when
and
. Consequently, deducing from Equation (
36), we obtain
Thus, there exists a function
such that
Therefore, from Equation (
35), we obtain
This can be expressed as
Theorem 4. The necessary and sufficient condition for the hypersurface of a Finsler space equipped with a Rander’s-type exponential-form metric defined in Equation (1) to be classified as a hyperplane of the second kind is delineated in Equation (37). Once more, Lemma 4, in conjunction with Equation (
25) and
, indicates that
does not form a hyperplane of the third kind. Therefore, the following holds.
Theorem 5. The hypersurface within a Finsler space , distinguished by a Rander’s-type exponential metric as specified in Equation (1), is incapable of being a hyperplane of the third kind. 6. Conclusions
Following an in-depth exploration of a distinct Finsler space defined by a Rander’s-type exponential-form metric with the expression , where represents the Riemannian metric and denotes the 1-form metric, this study has delved into the intrinsic properties of this specialized geometric space.
The research has primarily focused on investigating the behavior of hypersurfaces within this Finsler space and their resemblance to hyperplanes categorized into the first, second, and third kinds. By scrutinizing the conditions under which these hypersurfaces exhibit such characteristics, we have unveiled significant insights into the interplay between the metric structure and the geometric properties of the space.
Furthermore, our study has examined the reducibility of the Cartan tensor for these hypersurfaces in diverse forms, unveiling further layers of complexity within the geometric framework.
By shedding light on the nuanced relationship between the metric structure and the geometric features of Finsler spaces, this research contributes to a deeper understanding of the behavior of these spaces under specific conditions. The findings presented in this study pave the way for further exploration and research in the realm of Finsler geometry, offering a promising avenue for uncovering additional intriguing facets and applications within this field.