0

Let $M$ be a smooth manifold (could be $\mathbb{R}^n$) and $g_1(\cdot,\cdot),g_2(\cdot,\cdot)$ two inner products. Let $p \in M$ and denote by $\exp_1$ and $\exp_2$ the corresponding exponential maps based at $p$: $\exp_i: T_pM \to M$; both are local diffeomorphism mapping $0 \in T_pM$ to $p\in M$. Then $\exp_1^{-1}\circ \exp_2: T_pM \to T_pM$ is well-defined locally around $0$. Is anything known about this function? For example, is its Taylor series known?

user306330
  • 71
  • 5
  • 1
    hardly possible, if you don't know anything specific about $g_1,g_2$ – Simonsays Jun 26 '18 at 05:40
  • By non-degeneracy, you get a $(1,1)$-tensor field $A$ on $M$ such that $g_1(X,Y) = g_2 (AX,Y)$ for all $X,Y$. I'd think it would be reasonable to see what happens in terms of $A$, but as @Simonsays, you'd have to know more about $g_1$ and $g_2$ to try something concrete. – Ivo Terek Jun 26 '18 at 06:14

0 Answers0