Horoballs based at the same point in Hadamard manifolds

Question

In a Hadamard manifold if two horospheres based at a point $a$ are disjoint is it true that the distance between them is positive? If its not true in general what if the Hadamard manifold has pinched curvature.

Answer 1

By definition, a horosphere is the level set of a horofunction, $\{h(x)=c\}$. Two horoballs have the same base-point at infinity iff the corresponding horofunctions differ by a constant. Hence, we are talking about two horospheres $H_1=\{h(x)=c_1\}, H_2=\{h(x)=c_2\}$, $c_1< c_2$. Since $h$ is 1-Lipschitz, the minimal distance between $H_1, H_2$ is $\ge c_2-c_1$. (In fact, the minimal distance equals $|c_1-c_2|$. You can see this by considering the geodesic ray from a point on $H_2$ asymptotic to the point $a$, the basepoint of the horospheres.)