Properties of the Submanifold

Oct 13, 2025 by windailys.com 143 views

Introduction

In the realm of Riemannian geometry, a complete Riemannian manifold is a manifold that has a complete Riemannian metric, meaning that the distance between any two points in the manifold is finite. On the other hand, a closed Riemannian manifold is a manifold that is compact, meaning that it has a finite diameter. In this article, we will explore an example of a complete Riemannian metric and a connected embedded Riemannian submanifold that is complete but not closed.

Background

To understand the concept of a complete but not closed Riemannian submanifold, we need to recall some basic definitions from Riemannian geometry. A Riemannian manifold is a manifold equipped with a Riemannian metric, which is a smooth, symmetric, and positive-definite bilinear form on the tangent space of the manifold. A Riemannian metric induces a distance function on the manifold, which measures the distance between two points in the manifold.

A Riemannian manifold is said to be complete if the distance function is defined for all pairs of points in the manifold. In other words, the distance between any two points in the manifold is finite. A Riemannian manifold is said to be closed if it is compact, meaning that it has a finite diameter.

Example

Let's consider the following example. Let $M$ be the Euclidean space $\mathbb{R}^2$ equipped with the standard Euclidean metric. Let $P$ be the subset of $M$ consisting of all points $(x,y)$ such that $y = \sin(1/x)$ for $x \neq 0$ , and $y = 0$ for $x = 0$ . The set $P$ is a connected embedded Riemannian submanifold of $M$ .

The Riemannian metric on $P$ is induced from the standard Euclidean metric on $M$ . The distance function on $P$ is defined as the distance between two points in $P$ measured along the shortest path in $P$ .

Properties of the Submanifold

The submanifold $P$ has several interesting properties. First, it is complete because the distance function is defined for all pairs of points in $P$ . Second, it is not closed because it is not compact. In fact, the submanifold $P$ is not bounded, meaning that it does not have a finite diameter.

To see why $P$ is not bounded, consider the sequence of points $(1/n, \sin(1/n))$ for $n = 1, 2, 3, \ldots$ . This sequence converges to the point $(0, 0)$ in $P$ , but the distance between the points $(1/n, \sin(1/n))$ and $(0, 0)$ grows without bound as $n$ increases. Therefore, the submanifold $P$ is not bounded, and hence it is not closed.

Geometric Interpretation

The submanifold $P$ can be interpreted geometrically as a spiral curve in the Euclidean plane. The curve starts at the origin and spirals outward to infinity, never intersecting itself. The curve is complete because it has a finite distance between any two points, but it is not closed because it is not compact.

Conclusion

In this article, we have constructed an example of a complete Riemannian metric and a connected embedded Riemannian submanifold that is complete but not closed. The submanifold $P$ is a spiral curve in the Euclidean plane that starts at the origin and spirals outward to infinity. The submanifold $P$ is complete because it has a finite distance between any two points, but it is not closed because it is not compact.

References

[1] Lee, J. M. (2013). Riemannian manifolds: An introduction. Springer.
[2] O'Neill, B. (1983). Semi-Riemannian geometry: With applications to relativity. Academic Press.
[3] Spivak, M. (1979). A comprehensive introduction to differential geometry. Publish or Perish.

Appendix

In this appendix, we provide some additional information on the submanifold $P$ .

Distance Function

The distance function on $P$ is defined as the distance between two points in $P$ measured along the shortest path in $P$ . The distance function is given by the formula:

d((x_1, y_1), (x_2, y_2)) = \inf \left\{ \int_{\gamma} \sqrt{g_{ij} \dot{\gamma}^i \dot{\gamma}^j} \, dt \right\}

where $\gamma$ is a piecewise smooth curve in $P$ from $(x_1, y_1)$ to $(x_2, y_2)$ , and $g_{ij}$ is the Riemannian metric on $P$ .

Geodesics

A geodesic on $P$ is a curve that locally minimizes the distance function. The geodesics on $P$ are given by the equations:

\ddot{x} = 0, \quad \ddot{y} = -\frac{1}{x^2} \sin(1/x)

These equations can be solved to obtain the geodesics on $P$ .

Curvature

The curvature of $P$ is given by the formula:

K = \frac{1}{2} \left( \frac{\partial^2 g_{ij}}{\partial x^i \partial x^j} + \frac{\partial^2 g_{ij}}{\partial y^i \partial y^j} \right)

Q: What is a complete Riemannian manifold?

A: A complete Riemannian manifold is a manifold that has a complete Riemannian metric, meaning that the distance between any two points in the manifold is finite.

Q: What is a closed Riemannian manifold?

A: A closed Riemannian manifold is a manifold that is compact, meaning that it has a finite diameter.

Q: What is the difference between a complete and a closed Riemannian manifold?

A: A complete Riemannian manifold is a manifold that has a finite distance between any two points, while a closed Riemannian manifold is a manifold that has a finite diameter.

Q: Can a Riemannian manifold be both complete and closed?

A: Yes, a Riemannian manifold can be both complete and closed. In fact, many familiar manifolds, such as the Euclidean space and the sphere, are both complete and closed.

Q: Can a Riemannian manifold be complete but not closed?

A: Yes, a Riemannian manifold can be complete but not closed. In fact, the example we constructed in the previous article, the spiral curve in the Euclidean plane, is a complete but not closed Riemannian manifold.

Q: What are some examples of complete but not closed Riemannian manifolds?

A: Some examples of complete but not closed Riemannian manifolds include:

The spiral curve in the Euclidean plane, which we constructed in the previous article.
The hyperbolic plane, which is a complete but not closed Riemannian manifold.
The Lobachevsky plane, which is a complete but not closed Riemannian manifold.

Q: What are some applications of complete but not closed Riemannian manifolds?

A: Complete but not closed Riemannian manifolds have many applications in mathematics and physics, including:

Geometry and topology: Complete but not closed Riemannian manifolds are used to study the geometry and topology of manifolds.
Differential equations: Complete but not closed Riemannian manifolds are used to study differential equations on manifolds.
Relativity: Complete but not closed Riemannian manifolds are used to study the geometry of spacetime in relativity.

Q: How can I learn more about complete but not closed Riemannian manifolds?

A: There are many resources available to learn more about complete but not closed Riemannian manifolds, including:

Books: There are many books on Riemannian geometry and differential geometry that cover complete but not closed Riemannian manifolds.
Online courses: There are many online courses on Riemannian geometry and differential geometry that cover complete but not closed Riemannian manifolds.
Research papers: There are many research papers on complete but not closed Riemannian manifolds that can be found online.

Q: What are some common mistakes to avoid when working with complete but not closed Riemannian manifolds?

A: Some common mistakes to avoid when working with complete but not closed Riemannian manifolds include:

Assuming that a complete but not closed Riemannian manifold is closed.
Assuming that a complete but not closed Riemannian manifold is compact.
Failing to check the completeness and compactness of a Riemannian manifold.

Q: How can I apply complete but not closed Riemannian manifolds to real-world problems?

A: Complete but not closed Riemannian manifolds can be applied to many real-world problems, including:

Navigation: Complete but not closed Riemannian manifolds can be used to study the geometry of navigation systems.
Computer vision: Complete but not closed Riemannian manifolds can be used to study the geometry of images.
Robotics: Complete but not closed Riemannian manifolds can be used to study the geometry of robotic systems.

Q: What are some open problems in the study of complete but not closed Riemannian manifolds?

A: Some open problems in the study of complete but not closed Riemannian manifolds include:

The study of the geometry of complete but not closed Riemannian manifolds.
The study of the topology of complete but not closed Riemannian manifolds.
The study of the applications of complete but not closed Riemannian manifolds to real-world problems.