Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 115 / 4

For local coordinates $x^1,\dots ,x^n$, a curve is an affinely parametrized geodesic exactly when it satisfies the geodesic equationwhere the Christoffel symbols of the Levi-Civita connection are

For $v$ near $0\in T_{p}M$, let ${\gamma }_v$ be the unique geodesic with ${\gamma }_v(0)=p$ and ${\dot{\gamma }}_v(0)=v$. The exponential map is ${\exp }_p(v)={\gamma }_v(1)$.

The constant initial velocity $0$ gives ${\exp }_p(0)=p$. Varying the initial velocity through $sv$ yields ${\exp }_p(sv)={\gamma }_v(s)$, soThus $(d{\exp }_p{)}_0$ is the identity. The inverse function theorem makes ${\exp }_p$ a diffeomorphism from a neighborhood of $0$ onto a neighborhood of $p$. Coordinates from an orthonormal basis of $T_{p}M$ are the geodesic normal coordinates.

In geodesic coordinates built from an orthonormal basis, $g_{ij}(0)={\delta }_{ij}$. Every radial curve has coordinates $x(t)=ta$, and substitution into the geodesic equation givesTaking $t=1$ proves $x^i{x}^j{\Gamma }_{ij}^k(x)=0$.

Conversely, assume the metric and Christoffel-symbol conditions. For every $a$ in the star domain, $x(t)=ta$ satisfies the geodesic equation and has initial velocity $a$ in an orthonormal coordinate frame. Uniqueness of solutions to ordinary differential equations gives ${\varphi }^{-1}(ta)={\exp }_p(ta)$ wherever defined. The star-domain assumption covers all of $U$, so $\varphi$ is precisely a geodesic coordinate chart. This is the Radial Christoffel-symbol criterion for geodesic coordinates.

The metric induces an inner product on decomposable $r$-forms byextended bilinearly. On an oriented Riemannian manifold, the Hodge star operator is characterized byOn $r$-forms in dimension $n$,

Put $\alpha =\delta \eta$. The hypothesis says that the exact two-form $\beta =d\alpha$ is anti-self-dual. Since $M$ is compact without boundary, Stokes theorem givesHence $d\alpha =0$ by the exact anti-self-dual form on a compact four-manifold argument. Since $\delta$ is the formal $L^2$ adjoint of $d$,Therefore $\boxed{\delta \eta =0}$.