  Given a compact K\"ahler manifold $(X,\omega_0)$ let $\mathcal H_{0}$ be the
set of K\"ahler forms cohomologous to $\omega_0$. As observed by Mabuchi
\cite{m}, this space has the structure of an infinite dimensional Riemannian
manifold, if one identifies it with a totally geodesic subspace of $\mathcal
H$, the set of K\"ahler potentials of $\omega_0$. Following Donaldson's
research program, existence and regularity of geodesics in this space is of
fundamental interest. In this paper, supposing enough regularity of a geodesic
$u:[0,1]\to \mathcal H$, connecting $u_0 \in \mathcal H$ with $u_1 \in \mathcal
H$, we establish a Morse theoretic result relating the critical points of
$u_1-u_0$ to the critical points of $\dot u_0 = du/dt|_{t=0}$. As an
application of this result, we prove that on all K\"ahler manifolds, connecting
K\"ahler potentials with smooth geodesics is not possible in general. In
particular, in the case $X \neq \Bbb C P^1$, we will also prove that the set of
pairs of potentials that can not be connected with smooth geodesics has
nonempty interior. This is an improvement upon the findings of \cite{lv} and
\cite{dl}.
