Is the vector field associated with an element of the boundary at infinity on a Hadamard manifold smooth?

Question

A Hadamard manifold $M$ (complete, simply connected, non-positive sectional curvature) has a so-called boundary at infinity $\partial M$ whose elements are equivalence classes of unit-speed geodesic rays that are asymptotic i.e. two unit-speed geodesic rays $\gamma_1,\gamma_2:[0,\infty)\rightarrow 0$ are members of the same element of $\partial M$ iff $d(\gamma_1(t),\gamma_2(t))$ is bounded on the domain. For each $p\in M$ and $u\in \partial M$, there exists a unique element of $u$ associated with $p$ i.e. a unique unit geodesic ray $\gamma_p$ satisfying $\gamma_p(0)=p$ and $\gamma_p\in u$. Define a vector field $M\rightarrow TM$ by $p\rightarrow (p,\gamma_p'(0))$; that is, $p$ is mapped to the initial velocity of the element of $u$ associated with $p$. Is this vector field smooth?

I feel like this should be a known result but cannot find it in the literature. Any references would be appreciated.

Answer 1

This vector field is the negative of the gradient vector field of the/a Busemann function $b_u$ associated with $u$. (Below, I am assuming that the Riemannian metric is $C^\infty$.) The function is known to be $C^2$ for Hadamard manifolds but not $C^3$ (even for general Hadamard surfaces). See:

Heintze, Ernst; Im Hof, Hans-Christoph, Geometry of horospheres, J. Differ. Geom. 12, 481-491 (1977). ZBL0434.53038.

and

Ballmann, W.; Brin, M.; Burns, K., On the differentiability of horocycles and horocycle foliations, J. Differ. Geom. 26, 337-347 (1987). ZBL0609.53012.

From this, one sees that the vector field is $C^1$ but not, in general, $C^2$-smooth.

Ideally, these results belong to a textbook on Hadamard manifolds (that would summarize remarkable developments in the subject in 1970s and 1980s). Alas, such a book was never written, even though there was a couple of related books on general metric spaces of nonpositive curvature (Ballmann and Bridson-Haefliger) and several survey papers (as well as Eberlein's book and Gromov-Ballmann-Schroeder's book). But these were "close misses."