Morse-Bott critical manifold ($\ker {\mathrm{Hess}}_{x}f=T_{x}C$)

A Morse-Bott critical manifold is a smooth submanifold $C$ of critical points such that the kernel of the Hessian matrix at every $x\in C$ is exactly $T_{x}C$. The Hessian is therefore nondegenerate in normal directions. Repeated based great-circle geodesics on a round sphere form such critical manifolds, parametrized by their unit initial directions.