Killing jacobi fields from rotations

Given: The integer $n\ge 1$, the unit round sphere $S^n$ with its induced metric, the rotation generator $A$ and the rotation matrices $R_{\theta }$, and the field $X(x)=Ax$.

The unit sphere $S^n$ is a regular level set of $x\mapsto ⟨x,x⟩$, hence a smooth boundaryless $n$-manifold; at each $x\in S^n$ its tangent space is the kernel $x^{\perp }$ of the derivative, the inclusion is an immersion, and the restricted Euclidean inner product is the round Riemannian metric (A regular level set is an embedded submanifold, The tangent space of a regular level set is the kernel, Pullback of a riemannian metric is riemannian exactly for immersions, Riemannian metric and riemannian manifold, The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$).

On the first two coordinates the map $A$ is the matrix $\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$ and it annihilates the remaining coordinates, so $A^T=-A$, $A^2$ is the negative of the orthogonal projection onto $\mathrm{span}\{e_1,e_2\}$, and $A^3=-A$. Consequently $R_{\theta }=I+\sin \theta A+(1-\cos \theta )A^2,R_{\theta }^T=R_{-\theta },R_{\theta +\phi }=R_{\theta }{R}_{\phi },$ the last because $A$ commutes with $A^2$ and the trigonometric addition formulas hold; moreover $\frac{d}{d\theta }{R}_{\theta }=A{R}_{\theta }=R_{\theta }A,$ since $A^3=-A$ gives $A{R}_{\theta }=A+\sin \theta A^2-(1-\cos \theta )A=\cos \theta A+\sin \theta A^2$ (The derivatives of sine and cosine are cosine and minus sine, Parity and the Pythagorean identity for sine and cosine, algebra).

Each $R_{\theta }$ preserves the Euclidean inner product: $⟨R_{\theta }u,R_{\theta }v⟩=⟨u,v⟩$ for all $u,v\in {\mathbb{R}}^{n+1}$, because $R_{\theta }^T{R}_{\theta }=R_{-\theta }{R}_{\theta }=R_0=I$ by [F2]. In particular $\mathrm{\mid }{R}_{\theta }x\mathrm{\mid }=\mathrm{\mid }x\mathrm{\mid }$, so $R_{\theta }$ maps $S^n$ to itself (The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$, Parity and the Pythagorean identity for sine and cosine).

The curve $\theta \mapsto R_{\theta }x$ is the maximal integral curve of the field $X$ through $x$, so the local flow of $X$ is $\mathrm{\Phi }(\theta ,x)=R_{\theta }x$ (Local and global flows generated by a vector field, The derivatives of sine and cosine are cosine and minus sine).

A smooth local diffeomorphism $F$ of a Riemannian manifold is a (local) isometry when $F^{\ast }g=g$; for the sphere this means $g_{F(x)}(d{F}_{x}u,d{F}_{x}v)=g_x(u,v)$ for tangent vectors $u,v$ (Riemannian isometry and local isometry).

The Lie derivative of the metric along a vector field with flow $\mathrm{\Phi }$ is ${\mathcal{L}}_{X}g=\frac{d}{d\theta }{\mid }_{\theta =0}{\mathrm{\Phi }}_{\theta }^{\ast }g$, and a tensor field is flow-invariant exactly when its Lie derivative vanishes (The Lie derivative of a tensor field, A tensor field is flow-invariant exactly when its Lie derivative vanishes).

For a smooth vector field $X$ with ${\mathcal{L}}_{X}g=0$ (a Killing field in the sense of that proposition) and an affinely parametrized geodesic $\gamma$, the restriction $J(t)=X(\gamma (t))$ is a Jacobi field along $\gamma$ (Killing fields restrict to Jacobi fields along geodesics, Geodesic of an affine connection).

Every constant-speed parametrization of a great circle of the round sphere is a geodesic (Great circles as round-sphere geodesics).