Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 309 / 2

Use the curvature convention $[{\nabla }_b,{\nabla }_c]V^d=R^d{}_{abc}{V}^a$, consistent with the coordinate formula later in the paper. Choose normal coordinates at an arbitrary point $p$. The Levi-Civita connection has ${\Gamma }^d{}_{ab}(p)=0$ and is torsion free, so ${\Gamma }^d{}_{ab}={\Gamma }^d{}_{ba}$. At $p$,Hence the cyclic sum isEach derivative cancels using symmetry of the two lower connection indices. Because the Riemann curvature tensor is antisymmetric in its last two indices, full antisymmetrization over $a,b,c$ equals one third of this cyclic sum. ThusThe statement is tensorial and $p$ was arbitrary, so it holds in all coordinates everywhere. This is the first Bianchi identity.

Let $D_{abc}={\nabla }_a{\nabla }_b{K}_c$. Differentiating the Killing equation gives $D_{abc}=-D_{acb}$. For a covector, the Ricci identity isCall this difference $C_{abc}$. The differentiated Killing equation then givesTaking the first minus the second plus the third yieldsThe pair symmetries of the Riemann curvature tensor and the first Bianchi identity reduce the coefficient to $2R_{cbad}$. Therefore the second covariant derivative of a Killing vector is

For an affine parameter $\lambda$ on a geodesic, its tangent obeys $V^b{\nabla }_b{V}^a=0$. ThusThe second term vanishes by the geodesic equation; the first contracts a symmetric product $V^a{V}^b$ with the antisymmetric derivative from the Killing equation. This establishes the geodesic conserved quantity from a Killing vector:

In Cartesian coordinates on Euclidean space, the Levi-Civita connection and the Riemann curvature tensor vanish. The second covariant derivative of a Killing vector therefore givesEvery first partial derivative is constant on the connected space ${\mathbb{R}}^n$. Integrating once more gives $K^{\nu }=A^{\nu }+M_{\mu }{}^{\nu }{x}^{\mu }$, with constant coefficients. The Killing equation now becomesThus the coefficient is an antisymmetric matrix. Conversely, any such constant coefficients satisfy the Killing equation. The complete Euclidean Killing vector field is thereforeThe $n$ translation parameters and $n(n-1)/2$ rotation parameters give $n(n+1)/2$ independent Killing vector fields.

The index order in $M_{\mu }{}^{\nu }{x}^{\mu }$ fixes the rotation sign. With the entries specified in the paper, the integral curves of a vector field satisfyShift to $u=x^1-A^2$, $v=x^2+A^1$. Then $\dot{u}=v$, $\dot{v}=-u$, so $u^2+v^2$ is constant. If $x(0)=x_0$, put $u_0=x_0^1-A^2$ and $v_0=x_0^2+A^1$. Solving this linear ordinary differential equation gives the full local flow:The planar motion is a circle centred at $(A^2,-A^1)$, traversed clockwise as $\lambda$ increases. A nonzero vector $(A^3,\dots ,A^n)$ produces a helical orbit of a Euclidean Killing vector; zero drift gives a circle. At zero radius the curves are straight lines in the remaining directions, or stationary points if that drift also vanishes. These formulas are defined for every real $\lambda$.