Suppose $M$ is a Riemannian manifold and $f:M\to\mathbb{R}$ a differentiable function. I can form the Hessian $H$ of $f$ (with respect to the Levi-Civita connection); this is a symmetric bilinear form in the tangent bundle of $M$. The eigenspaces of $H$ fit together into distributions on $M$. My question is, under what conditions are these distributions integrable?
More precisely, assume that $\lambda\in\mathbb{R}$ is nowhere on $M$ an eigenvalue of $H$, then at every point of $M$ the tangent space splits into a direct sum of $T^{<\lambda}$ and $T^{>\lambda}$, where $T^{<\lambda}$ is the sum of the eigenspaces of $H$ whose corresponding eigenvalue is $<\lambda$ (and similarly for $T^{>\lambda}$). Under what conditions are $T^{<\lambda}$ and $T^{>\lambda}$ integrable distributions?
