Is the differential of the exponential map orthogonal? Let $M$ be a Riemannian manifold, and let $exp_x$ denote the exponential map at $x \in M$.  Assume the differential of the exponential map $D exp_x(v), v\in T_x M,$ is well defined.  Is $D exp_x(v)^* D exp_x(v) = Id$? If not, is there an explicit expression for $D exp_x(v)^* D exp_x(v)$?

No, this is true only for flat metrics.  Indeed, $T_xM$ is flat (it is just a vector space with the Riemannian metric associated to a certain inner product on the vector space), so if $D\exp_x$ were orthogonal on some open subset of $T_xM$, then $\exp_x$ would be locally an isometry on that open subset and so the image of that open subset in $M$ would be flat.
