 Solution

Use the curvature sign convention implicit in part (c): the Jacobi field equation is $D_t^{2}J+K_{\dot{\gamma }}J=0$, so the operator in a positively curved normal direction has positive eigenvalue. With the alternative convention $R_{\mathrm{std}}(X,Y)Z={\nabla }_X{\nabla }_{Y}Z-{\nabla }_Y{\nabla }_{X}Z-{\nabla }_{[X,Y]}Z$ used for the catalog's Jacobi curvature operator, the paper's tensor is $R=-R_{\mathrm{std}}$, and its operator is $K_v(x)=R_{\mathrm{std}}(x,v)v$. This fixes the sign throughout Q1.

Let $r(A,B,C,D)=⟨R(A,B)C,D⟩$. Pair interchange for the Riemann curvature tensor givesThus $K_v$ is self-adjoint with respect to the Riemannian metric. The curvature sign does not affect this conclusion. The spectral theorem for real symmetric matrices therefore supplies the real orthonormal eigenbasis used in part (b). Also $K_{v}v=0$, so the tangent direction is a zero-eigenvalue direction.