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Suppose we have a compact smooth riemannian manifold $(M,g)$ and a $\mathcal{C}^2$ diffeomorphism $f$ of this manifold. I am studying the mapping

$$\tilde{f}(x)= \exp_{f(x)}^{-1} \circ f \circ \exp_x \colon U_x \subset T_x(M) \to T_{f(x)}(M) $$

In particular, I am trying to understand the domain of definiton of this mapping. The question is the following:

What is the "maximal" neighbourhood in which I can define the map $\tilde{f}$ ? In particular, how does this neighbourhood depend on the choice of the riemannian metric, on the choice of $f$ and on the choice of the point $x \in U$. Thanks!

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  • $\begingroup$ It sounds like you are interested in the radius of injectivity for the exponential map? This is a topic of many introductory differential geometry textbooks. $\endgroup$
    – Ryan Budney
    Commented Nov 6, 2021 at 18:16
  • $\begingroup$ Why is knowing the injectivity radius enough? I think I also need to take into account how to deal with the domain of the inverse. In other words, I think I need also the domain of definition of $\exp_{x}^{-1}$ $\endgroup$
    – Giuseppe Tenaglia
    Commented Nov 6, 2021 at 18:32
  • $\begingroup$ I suppose I view the injectivity radius as telling you the maximal radius open ball in the domain of $exp_x^{-1}$. $\endgroup$
    – Ryan Budney
    Commented Nov 6, 2021 at 18:36
  • $\begingroup$ If I do not have a distance in the manifold, how can I consider the concept of radius? $\endgroup$
    – Giuseppe Tenaglia
    Commented Nov 6, 2021 at 18:46
  • $\begingroup$ Do you know any introductory book for differential geometry? I am extremely new to the topic $\endgroup$
    – Giuseppe Tenaglia
    Commented Nov 6, 2021 at 18:46

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