Questions about the differential

[mathwonk]

be aware Lavinia is talking about something slightly different from a map M-->Tp(M). She is talking about maps M-->T(M), without the fixed subscript p.

I.e. a vector field on M is a map v:M-->T(M) such that for each p in M, v(p) belongs to Tp(M). So the p is varying in a vector field.

 

[orion]

Thanks, everyone. That clears up a lot of issues.

I actually did come across the exponential map in one set of notes I found on the internet, but I put it aside until I get the basics down.

lavinia, I neglected to thank you for your post earlier. Thanks.

 

[lavinia]

I don't think we have discussed maps like that. If M is equipped with a metric, then the "exponential map" at p takes an arbitrary $v\in T_pM$ to $\gamma(1)\in M$, where $\gamma$ is the unique geodesic through p whose tangent vector at p is v. I doubt that the exponential map is always invertible, but I would guess that if it isn't, it probably has a restriction that's invertible.

The exponential map is always locally invertible. Its differential is non-singular at the origin of the tangent space being exponentiated.

The example of the sphere that Mathwonk described shows that the exponential map can have singularities and in general it does. The study of its singularities - so called conjugate points - is the subject of Morse Theory.

In some cases, the exponential map can have no singularities but still not be invertible. This is true for instance of compact flat Riemannian manifolds since on them, the Riemann curvature tensor is identically zero so there are no conjugate points.For instance, for the flat torus, the exponential map is a covering projection.