Riemannian Geometry exponential map and distance

In summary: Yes, you can use any connection on the tangent bundle. The important thing is that the connection be left or right invariant.
  • #1
ireallymetal
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Hi all, I was wondering what the relationship between the Riemannian Geometry exponential map and the regular manifold exponential map and for the reason behind the name.
 
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  • #2
Thanks for the post! Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
 
  • #3
In a manifold with a connection, the exponential map is defined by using the geodesics of that connection (geodesic defined here as a curve for which the derivative of its tangent vector field in the direction of the curve is zero). In a Riemannian manifold, you use a particular connection for this construction, the Levi-Civita connection.

The name 'exponential map' is because this map generalizes the notion of exponential function and of the exponential of a matrix. This can be seen in the context of Lie groups. A Lie group is a group that is also a differentiable manifold for which the group operations are smooth. The tangent space at the identity of the group is its Lie algebra. So, we can build the exponential map which goes from the Lie algebra to the group. The connection is taken to be left or right invariant. The geodesics in this case are the usual integral curves of the left or right invariant vector fields.

If the group is the real numbers under the multiplication, then the algebra is the real numbers under the sum, and, in this case, the exponential map simply reduces to the usual exponential function.

For matrix Lie groups (i.e., groups whose elements are nxn matrices), the exponential map reduces to the usual matrix exponential.
 
  • #4
ireallymetal said:
Hi all, I was wondering what the relationship between the Riemannian Geometry exponential map and the regular manifold exponential map and for the reason behind the name.
What do you mean by "regular manifold exponential map"?
 
  • #5
aleazk said:
In a manifold with a connection, the exponential map is defined by using the geodesics of that connection (geodesic defined here as a curve for which the derivative of its tangent vector field in the direction of the curve is zero). In a Riemannian manifold, you use a particular connection for this construction, the Levi-Civita connection

Can you explain the exponential map for a connection on the tangent bundle that is not Levi-Civita?
 

1. What is the Riemannian Geometry exponential map?

The Riemannian Geometry exponential map is a mathematical tool used to describe the relationship between a point on a Riemannian manifold and a geodesic curve passing through that point. It maps a point on the manifold to a point on the tangent space at that point, allowing for the calculation of distances and angles on the manifold.

2. How is the Riemannian Geometry exponential map calculated?

The Riemannian Geometry exponential map is typically calculated using a Taylor series expansion, which approximates the map as a polynomial function. This allows for the efficient computation of the map and its inverse, the logarithmic map.

3. What is the significance of the Riemannian distance function?

The Riemannian distance function, also known as the intrinsic distance, is a measure of the length of the shortest path, or geodesic, between two points on a Riemannian manifold. It is a fundamental concept in Riemannian Geometry as it allows for the quantification of distances and the definition of curvature on the manifold.

4. How is the Riemannian distance function related to the exponential map?

The Riemannian distance function can be calculated using the exponential map. By taking the logarithm of the image of a point under the exponential map, the distance between that point and the origin on the tangent space can be determined. This distance corresponds to the geodesic distance between the original point and the point on the manifold.

5. What are some applications of the Riemannian Geometry exponential map and distance?

The Riemannian Geometry exponential map and distance have various applications in fields such as physics, engineering, and computer science. They are used in the study of curved spaces, such as in general relativity, and in the development of algorithms for optimization problems and machine learning. They also have practical applications in fields like computer graphics and medical imaging.

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