To let $ M $ a riemannian manifold n-dimensional without boundary. Show that $$ langle exp ^ {- 1} _ {x} (y), exp ^ {- 1} _ {x} (y) rangle_ {g_ {x}} = langle exp ^ {- 1 } _ {y} (x), exp ^ {- 1} _ {y} (x) rangle_ {g_ {y}} qquad, forall y in mathcal {U} $$

Where $ mathcal {U} $ is a normal neighborhood of $ x in M $

My approach: Let $ mathcal {U} $ a normal neighborhood of $ x in M $that contain one $ y in M $, If $ c (t) $ is the geodesic curve connecting both points ($ x, y $), i. $ c (0) = x $ and $ c (1) = y $, then $ c ^ { prime} (0) = exp_ {x} ^ {- 1} (y) $,

Now consider the parallel transport of $ c ^ { prime} (0) $ along $ c $ (Note that $ c $ is the geodesic curve), then we have $$ langle c ^ { prime} (0), c ^ { prime} (0) rangle_ {g_ {x}} = langle c ^ { prime} (1), c ^ { prime} (1) rangle_ {g_ {y}} $$

I do not know if my proof is correct *. However, why, if we take the parallel transport, $ P_ {c} rvert_ {0} ^ {1} $ along the geodesic line, the end point coincides with the point $ c (1) = y $? Is this point unique?

Every idea is appreciated. THANK YOU SO MUCH!