Suppose $M$ is a finite dimensional $C^\infty$-manifold and $\nabla$ is an affine connection on $M$, we know that exponential mapping and logarithm mapping are well-defined locally. Fix a point $p_0\in M$, $\log_{p_0}(\cdot):M\to T_{p_0} M$ is well-defined on a neighborhood of $p_0$. Now my questions are:
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Is $\log_{p_0}(\cdot)$ differentiable or differentiable almost everywhere? If it is, is there a closed form?
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Intuitively, I guess that it's the parallel transport from $T_p M$ to $T_{p_0}M$, where $p$ represents the argument. Is this correct in general? Does it hold in any special cases?
So far, I mainly focus on two cases, but I welcome any more general results.
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$M$ is a Riemannian manifold and $\nabla$ is the Levi-Civita connection.
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$M$ is a Lie group and $\nabla$ represents the canonical left-invariant connection, so the exponential mapping is the usual one on Lie group.
Please enlighten me with any related results, examples or references. Thank you!
Best Answer
First a note on the logarithmic map, which isn't really a single map at all, but a family of local inverses of the exponential map.
The exponential map $\exp_{p_0}$ as full rank at $0\in T_{p_0}M$, so by inverse function theorem there are open sets $\mathcal{U}\subseteq T_{p_0}M$, $\mathcal{V}\subseteq M$ such that $\exp_{p_0}:\mathcal{U}\to\mathcal{V}$ is a diffeomorphism. There isn't a canonical choice of neighborhood $\mathcal{U}$. Having chosen such an appropriate $\mathcal{U}$, there is a corresponding local inverse $\log_{p_0}:\mathcal{V}\to\mathcal{U}$ which is certainly differentiable on its domain, and it's differential at every point is the inverse of $d\exp_{p_0}$. It isn't possible to extend this local inverse to a global one, however, unless $\exp_p$ is a diffeomorphism.
Fixing a particular choice of domain $\mathcal{U}$, we can compare the two maps you're interested in by looking at their inverses. Fix $v\in\mathcal{U}$, and let $p=\exp_{p_0}(v)$.
For simplicity, I'll identify $T_vT_{p_0}M$ with $T_{p_0}M$ in the standard way. Rather that look at $d_p\log_{p_0}$, it is simpler to consider its inverse $d_v\exp_{p_0}:T_{p_0}M\to T_p M$. We can equivalently compare this to the parallel transportation $F:T_{p_0}M\to T_pM$ along the the geodesic $\gamma$ starting at $v$. These two maps are not generally the same. A detailed comparison between them is made in this mathoverflow post. (Though only in the Riemannian case; I suspect the presence of torsion will show up in the Jacobi equation.)
As an alternative intuition for the problem, one can think of $d_v\exp_{p_0}$ as describing how an infinitesimal change in the initial velocity of a geodesic causes its endpoint to move. $d_{p}\log_{p_0}$ is then the inverse, it takes an infinitesimal displacement of the endpoint and determines what change in velocity would bring about this displacement. This can be made precise using Jacobi fields.