= Solution
Use the curvature sign convention implicit in part (c): the <Jacobi field> equation is $D_t^2J+K_{\dot\gamma}J=0$, so the operator in a positively curved normal direction has positive <eigenvalue>. With the alternative convention $R_{\mathrm{std}}(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z$ 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 $r(A,B,C,D)=\langle R(A,B)C,D\rangle$. Pair interchange for the <Riemann curvature tensor> gives
$$
\langle K_vx,y\rangle=r(v,x,v,y)=r(v,y,v,x)=\langle x,K_vy\rangle.
$$
Thus \b[$K_v$ 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 $K_vv=0$, so the tangent direction is a zero-eigenvalue direction.
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