Let $(M,g)$ be a smooth Riemannian manifold and let $\gamma:[a,b]\to M$ be a geodesic. Assume that $\gamma(b)$ is not conjugate to $\gamma(a)$ along $\gamma$. Let $\mathcal V_0(\gamma)$ be the real [vector space](/page/Vector%20Space) of piecewise smooth vector fields $V$ along $\gamma$ satisfying $V(a)=V(b)=0$, and let $I_\gamma:\mathcal V_0(\gamma)\times \mathcal V_0(\gamma)\to \mathbb R$ be the index form

\begin{align*} I_\gamma(V,W) = \int_a^b \left( g_{\gamma(t)}\left(\nabla_t V(t),\nabla_t W(t)\right) - g_{\gamma(t)}\left(R_{\gamma(t)}(V(t),\dot\gamma(t))\dot\gamma(t),W(t)\right) \right) \,d\mathcal L^1(t), \end{align*}

where $\nabla_t$ denotes covariant differentiation along $\gamma$ and $R$ is the Riemann curvature tensor with convention $R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z$.

For $c\in(a,b)$, let

\begin{align*} m(c) = \dim\{J:\,[a,c]\to TM \text{ a Jacobi field along }\gamma|_{[a,c]} \mid J(a)=0,\ J(c)=0\}. \end{align*}

Then

\begin{align*} \operatorname{ind}(I_\gamma) = \sum_{c\in(a,b)} m(c), \end{align*}

where $\operatorname{ind}(I_\gamma)$ is the maximal dimension of a subspace of $\mathcal V_0(\gamma)$ on which $I_\gamma$ is negative definite. The sum is finite.