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.

Related Differential Geometry News on Phys.org

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

Gold Member
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

Gold Member
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

Gold Member
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?

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving