Some relations on Hadamard manifolds

by user678879   Last Updated August 14, 2019 09:20 AM - source

I'm new to Hadamard manifolds. For some purposes in mathematical analysis, I need to know if the following relations are true in Hadamard manifolds. Here $M $ is a Hadamard manifold and $\exp_p^{-1}:M\to T_pM $ and $P^{xp}:T_xM\to T_pM $ is the parallel transport.

$$\exp_p^{-1}p=o$$ $$P^{xp}\exp_x^{-1}p=-\exp_p^{-1}x$$ $$\exp_p^{-1}x+P^{xp}\exp_x^{-1}y=\exp_p^{-1}y$$

These relations seems to be true in $\mathbb R^n $. Since we have $\exp_p^{-1}x=x-p $ there and the proof is very straightforward. But in order to prove them in Hadamard manifolds, I need to have some start point, Could anyone help me, please?

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