Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 16 / 1

Write $T=\dot{\gamma }$ along an affinely parametrized geodesic and $D_t={\nabla }_T$ for the covariant derivative from the Levi-Civita connection. We use the Riemann curvature tensor convention $R(X,Y)Z={\nabla }_X{\nabla }_{Y}Z-{\nabla }_Y{\nabla }_{X}Z-{\nabla }_{[X,Y]}Z$, for which positive sectional curvature means $⟨R(X,Y)Y,X⟩>0$ on an orthonormal pair. A Jacobi field is a smooth vector field along $\gamma$ satisfyingSuch a field describes the first-order separation of nearby geodesics, as the geodesic variation construction in part (c) shows.

For $t_0>0$, $\gamma (t_0)$ is a conjugate point of $\gamma (0)$ along this geodesic if a nonzero Jacobi field satisfies $J(0)=J(t_0)=0$. The multiplicity of a conjugate point isIt is equivalently the dimension of the kernel of the differential of the Riemannian exponential map at $t_0\dot{\gamma }(0)$, by Jacobi fields vanishing at their initial point. Conjugacy concerns the specified geodesic and its two parameter values, even when its endpoints represent the same point of the Riemannian manifold.

Suppose $J$ is a Jacobi field vanishing at both endpoints of a nonconstant geodesic segment $[0,t_0]$. Its tangential component $h(t)=⟨J(t),T(t)⟩$ satisfiesbecause $D_{t}T=0$ and the Riemann curvature tensor has the usual skew-adjoint symmetry. The two zero endpoint values imply $h\equiv 0$. Both $J$ and $D_{t}J$ are therefore perpendicular to $T$.

The linear map from these Jacobi fields to $T(0{)}^{\perp }$ given by $J\mapsto D_{t}J(0)$ is injective: $J(0)=0$ together with $D_{t}J(0)=0$ forces $J\equiv 0$ by uniqueness for the linear ordinary differential equation defining a Jacobi field. Its target has dimension $n-1$, soThis proves the endpoint-vanishing normal Jacobi fields have dimension at most n-1 bound. For a constant geodesic the equation is $J^{\prime \prime }=0$, and two zero values instead force $J=0$, so there is no conjugate point.

Put $p=\gamma (0)$, $v=\dot{\gamma }(0)$, $a=J(0)$ and $b=D_{t}J(0)$. Choose the short curve $c(s)={\exp }_p(sa)$ and let $P_s:T_{p}M\to T_{c(s)}M$ be parallel transport along $c$. The tangent vectorssatisfy $V(0)=v$ and ${\nabla }_{s}V(0)=b$. Define a geodesic variation byFor each fixed $s$ this is an affinely parametrized geodesic with initial point $c(s)$ and initial velocity $V(s)$. Although the Riemannian manifold need not be complete, smooth dependence for the geodesic equation on a neighborhood of the compact reference segment provides a single $\epsilon >0$ on which $F$ exists for all $|s|<\epsilon$ and $0\le t\le 1$. Thus completeness is unnecessary.

Set $X(t)={\partial }_{s}F(0,t)$. The torsion-free Levi-Civita connection gives ${\nabla }_s{\partial }_{t}F={\nabla }_t{\partial }_{s}F$. Differentiating ${\nabla }_t{\partial }_{t}F=0$ with respect to $s$, and commuting the two covariant derivatives, yieldsAlso $X(0)=c^{\prime }(0)=a$ and $D_{t}X(0)={\nabla }_{s}V(0)=b$. Uniqueness for the Jacobi field equation now gives $X(t)=J(t)$ on the whole segment. This is realization of Jacobi fields by geodesic variations.

The printed formula switches the order of its two variables. The consistent notation here is $F(0,t)=\gamma (t)$ and $J(t)={\partial }_{s}F(0,t)$. A parametrized surface here means a smooth variation map; it cannot be required to be an immersion, since the allowed Jacobi field $J=0$ would already violate that condition along the reference curve.