How to prove that $\forall$ Riemannian manifold $M$, $T\in C^{1}(M)$ and $x,y\in M$ close enough: $$d(Ty,\exp_{Tx}(D_xT\ \exp^{-1}_xy))\leq d(x,y)$$
My attempts so far were only able to show the inequality up to a multiplicative constant depending on $T$, which is not good enough. This should be true as it appears in the book "Smooth ergodic theory for endomorphisms", and is true for the euclidean case.
*I was able to show the requested result under the assumption that there exists a small enough *neighborhood $U$ of $x$ such that, it can be mapped isometrically to $\mathbb{R}^{n}$, or some neighborhood in it *($dimM=n$) with normal euclidean metric- is that statement true?
-I now know that the suggested statement with *'s infront of the lines is incorrect.
Any help would be very much appreciated.