# Some relations on Hadamard manifolds

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?