Complete riemannian metric

In summary, any smooth manifold can be given a complete Riemannian metric, as each point on the manifold can be locally represented as Rn which can then be given a metric. However, it is not clear what is meant by a "complete" metric, and it is not specified how to find such a metric for a given manifold.
  • #1
mtiano
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Is it true that any smooth manifold admits a complete riemannian metric? Can you prove it? If not can you give a counter example? Obviously we can always put a riemannian metric on any smmoth manifold the question is does the differentable structure allow us to find a complete one.
 
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  • #2
mtiano said:
Is it true that any smooth manifold admits a complete riemannian metric? Can you prove it? If not can you give a counter example? Obviously we can always put a riemannian metric on any smmoth manifold the question is does the differentable structure allow us to find a complete one.
Although I'm not 100 % sure on what you mean by the qualifier "complete" but any smooth manifold can be given more structure by adding a metric to the space. To see this recall that every point on a manifold looks locally like Rn which can always be given a metric. So when you specify the distance relationship you wish to to all points on the manifold then you've specified a metric for that manifold.

Best wishes

Pete
 
  • #3


A complete Riemannian metric on a smooth manifold is a metric that is defined on the entire manifold and has no "incomplete" points, meaning that any two points on the manifold can be connected by a geodesic path of finite length.

It is not necessarily true that any smooth manifold admits a complete Riemannian metric. There are counterexamples, such as the real line with the standard Euclidean metric. This metric is not complete, as there are points on the real line (such as infinity) that cannot be reached by a finite geodesic.

However, there are some conditions under which a smooth manifold is guaranteed to admit a complete Riemannian metric. For example, if the manifold is compact, meaning that it is closed and bounded, then it is always possible to find a complete Riemannian metric on the manifold. This is a consequence of the Hopf-Rinow theorem, which states that any compact Riemannian manifold is geodesically complete.

Another condition that guarantees the existence of a complete Riemannian metric on a smooth manifold is if the manifold is simply connected. In this case, the manifold is topologically equivalent to Euclidean space, and it is always possible to construct a complete Riemannian metric on Euclidean space.

In general, the existence of a complete Riemannian metric on a smooth manifold is not guaranteed, but there are certain conditions under which it is possible to construct one.
 

1. What is a Riemannian metric?

A Riemannian metric is a mathematical tool that is used to measure distances and angles on a curved surface, such as a manifold. It is a function that assigns a length to each tangent vector at each point on the surface, allowing for calculations of lengths and angles between vectors.

2. How is a Riemannian metric defined?

A Riemannian metric is defined as a positive definite inner product on the tangent space at each point of a manifold. This means that for any two tangent vectors, the metric assigns a positive number, representing the inner product between the two vectors, which can be interpreted as the length of one vector projected onto the other.

3. What is the significance of a complete Riemannian metric?

A complete Riemannian metric is one that can measure distances between any two points on a manifold. This is important because it allows for the calculation of geodesics, which are the shortest paths between points on a curved surface. A complete metric also ensures that the manifold is connected, meaning that any two points can be connected by a smooth curve.

4. How is a complete Riemannian metric different from an incomplete one?

An incomplete Riemannian metric cannot measure distances between all points on a manifold. This means that there are some points that cannot be connected by a smooth curve, making the manifold disconnected. Incomplete metrics can arise from certain types of singularities or when the manifold has a boundary.

5. What are some applications of complete Riemannian metrics?

Riemannian metrics have many applications in mathematics and physics. They are used in the study of differential geometry, general relativity, and optimization problems. They also have applications in computer science, such as in image and signal processing, as well as in machine learning and data analysis.

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