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Let $M$ be a Riemannian Manifold, for each $p \in M$ we will denote $\exp_p : U_p \rightarrow M$ the exponential funtion, beeing $U_p \subseteq T_p(M)$ a small enough open connected set with $0 \in U_p$. When I say "small enough" I mean small enough for the exponential funtion to be well defined, to be a difeomorphism with it's image, and small enough for my conjecture to hold (if there even exists such an open set).

Before stating my conjecture, let's denote $\log_p : \exp_p{(U_p)} \rightarrow U_p \subseteq T_p{(M)}$ the inverse function of the exponential map over $p$. My conjecture is that if $p,q \in M$ are such that $V_{p,q} :=\exp_p{(U_p)} \cap \exp_q{(U_q)} \neq \emptyset$ (for simplicity, we can assume $V_{p,q}$ is connected) then the funtion $$\log_q \circ \exp_p : \exp_p^{-1}(V_{p,q}) \rightarrow \exp_q^{-1}(V_{p,q})$$

Is an affine linear function (wich I'm going to explain what it means to me).

  • An function $f : \Bbb R^n \rightarrow \Bbb R^m$ is an affine linear function if $g(x):=f(x)-v$ is a linear function for some $v \in \Bbb R^m$
  • If $U \subseteq \Bbb R^n$ is an open connected set, a function $h : U \rightarrow \Bbb R^m$ is affine linear if there exists an affine linear function $f : \Bbb R^n \rightarrow \Bbb R^m$ such that ${f|}_U =h$
  • If $U \subseteq \Bbb R^n$ is an open set, we say that $h : U \rightarrow \Bbb R^m$ is affine linear if it's affine linear restricted to every connected component of $U$
  • Finally, let $U \subseteq \Bbb R^n$ and $V \subseteq \Bbb R^m$ be open sets, we say that $h : U \rightarrow V$ is affine linear if $i \circ h : U \rightarrow \Bbb R^m$ is affine linear, beeing $i : V \rightarrow \Bbb R^m$ the inclusion map.

The same definitions holds for arbitrary finite dimensional (real) vector spaces. I've seen many examples in wich this happens to be true so I was wondering if is a general thing. Also, for some reason I belive the parallel transport will play an important role (taking the Levi-Civita connection)

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This is very far from being true. Normal coordinate charts are not as well adapted to a Rimannian metric as one may think. They only behave well with respect to geodesics through the point at with the normal coordinate chart is centered. Suppose that the conjecture were true and consider a normal coordinate chart centered at $p$. Then we can apply the conjecture to conclude that all geodesics that meet this chart are given by affine lines in that chart (since affine functions map affine lines to affine lines). This would even be true including parametrizations. But even if one requires that there are local charts in which the all geodescis are straight lineas as unparametrized curves, this implies a condition known as "projective flatness" of the Levi-Civita connection. By a classical theorem of Beltrami, a Riemannian manifold has projectively flat Levi-Civita connection if and only if it has constant sectional curvature. (And even in this case, you don't get the "right" parametrizations in general.) So I suppose that in the form you expect the condition only holds for flat metrics.

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