Riemannian Geometry exponential map and distance

[ireallymetal]

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.

 

[Greg Bernhardt]

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?

 

[aleazk]

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.

 

[lavinia]

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"?

 

[lavinia]

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?