Convexity of the Riemannian Distance

Oct 13, 2025 by windailys.com 20 views

Introduction

In the realm of Riemannian geometry, the concept of Riemannian distance plays a crucial role in understanding the properties of a Riemannian manifold. The Riemannian distance, denoted by $d$ , is a metric that measures the distance between two points on a Riemannian manifold. In this article, we will explore the relationship between the convexity of the Riemannian distance and its non-degeneracy. Specifically, we will investigate the condition under which the Riemannian distance is non-degenerate, given that it is convex.

Convexity of the Riemannian Distance

Let $f(t) = d(p, \gamma(t))$ , where $d$ is the Riemannian distance and $\gamma: [0,1] \to M$ is a geodesic connecting $p$ and $q$ on a Riemannian manifold $M$ . A function $f$ is said to be convex if for all $t \in [0,1]$ , we have

f(t) \leq \frac{1}{2} f(0) + \frac{1}{2} f(1).

In other words, the function $f$ lies below its chord. Geometrically, this means that the graph of $f$ is a convex function.

Non-Degeneracy of the Riemannian Distance

The Riemannian distance $d$ is said to be non-degenerate if for any two points $p$ and $q$ on the manifold $M$ , the distance $d(p,q)$ is strictly positive. In other words, the distance between any two points is always greater than zero.

Relationship between Convexity and Non-Degeneracy

We now investigate the relationship between the convexity of the Riemannian distance and its non-degeneracy. Specifically, we want to show that if the Riemannian distance is convex, then it is non-degenerate.

Theorem

Let $M$ be a Riemannian manifold and $d$ be the Riemannian distance on $M$ . Suppose that for any two points $p$ and $q$ on $M$ , the function $f(t) = d(p, \gamma(t))$ is convex, where $\gamma: [0,1] \to M$ is a geodesic connecting $p$ and $q$ . Then, the Riemannian distance $d$ is non-degenerate.

Proof

Let $p$ and $q$ be any two points on the manifold $M$ . Suppose, for the sake of contradiction, that the distance $d(p,q)$ is zero. Then, the function $f(t) = d(p, \gamma(t))$ is identically zero, since $\gamma$ is a geodesic connecting $p$ and $q$ .

However, this contradicts the convexity of the function $f$ , since a convex function cannot be identically zero. Therefore, our assumption that the distance $d(p,q)$ is zero must be false, and we conclude that the Riemannian distance $d$ is non-degenerate.

Geometric Interpretation

The result we have just proved has a beautiful geometric interpretation. Suppose that the Riemannian distance $d$ is convex, and let $p$ and $q$ be any two points on the manifold $M$ . Then, the function $f(t) = d(p, \gamma(t))$ is convex, and its graph lies below its chord.

Geometrically, this means that the distance between $p$ and $q$ is always greater than the distance between $p$ and any point on the geodesic $\gamma$ between $p$ and $q$ . In other words, the distance between $p$ and $q$ is always greater than the distance between $p$ and any point on the "shortest path" between $p$ and $q$ .

Conclusion

In this article, we have investigated the relationship between the convexity of the Riemannian distance and its non-degeneracy. Specifically, we have shown that if the Riemannian distance is convex, then it is non-degenerate. This result has a beautiful geometric interpretation, and it provides a new insight into the properties of Riemannian manifolds.

References

[1] Riemannian Geometry by M. Spivak, Publish or Perish, Inc.
[2] Convex Geometry by A. Barvinok, Cambridge University Press.
[3] Metric Spaces by M. Deza, Springer-Verlag.

Appendix

In this appendix, we provide a brief overview of the necessary background material on Riemannian geometry and convex geometry.

Riemannian Geometry

A Riemannian manifold is a smooth manifold equipped with a Riemannian metric, which is a positive-definite symmetric bilinear form on the tangent space at each point. The Riemannian distance is a metric that measures the distance between two points on the manifold.

Convex Geometry

Convex geometry is the study of convex sets and convex functions. A convex set is a set that contains all the line segments connecting any two points in the set. A convex function is a function that lies below its chord.

Metric Spaces

Introduction

In our previous article, we explored the relationship between the convexity of the Riemannian distance and its non-degeneracy. Specifically, we showed that if the Riemannian distance is convex, then it is non-degenerate. In this article, we will answer some frequently asked questions about this result.

Q: What is the Riemannian distance?

A: The Riemannian distance is a metric that measures the distance between two points on a Riemannian manifold. It is a fundamental concept in Riemannian geometry and is used to study the properties of Riemannian manifolds.

Q: What is convexity?

A: Convexity is a property of a function that means it lies below its chord. In other words, a convex function is a function that is always greater than or equal to the average of its values at any two points.

Q: What is non-degeneracy?

A: Non-degeneracy is a property of a metric that means it is always strictly positive. In other words, a non-degenerate metric is a metric that is always greater than zero.

Q: Why is convexity important in Riemannian geometry?

A: Convexity is important in Riemannian geometry because it provides a way to study the properties of Riemannian manifolds. Specifically, convexity is used to study the behavior of geodesics, which are the shortest paths on a Riemannian manifold.

Q: Can you give an example of a Riemannian manifold with a convex Riemannian distance?

A: Yes, an example of a Riemannian manifold with a convex Riemannian distance is the Euclidean space. The Euclidean space is a Riemannian manifold with a flat metric, and the Riemannian distance is convex.

Q: Can you give an example of a Riemannian manifold with a non-degenerate Riemannian distance?

A: Yes, an example of a Riemannian manifold with a non-degenerate Riemannian distance is the sphere. The sphere is a Riemannian manifold with a curved metric, and the Riemannian distance is non-degenerate.

Q: What are the implications of this result?

A: The implications of this result are far-reaching. Specifically, this result provides a new way to study the properties of Riemannian manifolds. It also provides a new tool for studying the behavior of geodesics on Riemannian manifolds.

Q: Can you provide a proof of this result?

A: Yes, a proof of this result can be found in our previous article. The proof is based on the definition of convexity and the properties of Riemannian manifolds.

Q: What are some open problems related to this result?

A: Some open problems related to this result include:

Can we generalize this result to higher-dimensional Riemannian manifolds?
Can we find a more general condition for convexity that is not dependent on the Riemannian distance?
Can we use this result to study the properties of other types of manifolds, such as complex manifolds?

Conclusion

In this article, we have answered some frequently asked questions about the relationship between the convexity of the Riemannian distance and its non-degeneracy. We hope that this article has provided a useful resource for those interested in Riemannian geometry and convex geometry.