Math 6510 • Differentiable Manifolds

Let \(M\subset \mathbb {R}^{n+1}\) be the graph \(x^{n+1}=f(x^1,\ldots ,x^n)\) where \(f\in C^\infty (\Omega )\) and \(\Omega \subset \mathbb {R}^n\) is a convex open set. Suppose the Hessian of \(f\) has constant rank i.e. for some \(k\) such that \(0\le k<n\) and every \(x\in \Omega \), \[ \operatorname {rank}\left (\frac {\partial ^2 f(x)}{\partial x^i\partial x^j}\right )=k. \] Show that \(M\) is foliated by \((n-k)\) dimensional manifolds whose tangent vectors lie in \(\ker (dG)\) where \(G:M\to \mathbb {R}^n\) is given by \[ G(x)=\left (\frac {\partial f}{\partial x^1},\ldots , \frac {\partial f}{\partial x^n}\right ). \] Show that a connected component \(L\) of such a “leaf” manifold locally coincides with points where \(G\) is constant. In fact, the graph of \(L\) is an affine subspace of \(\mathbb {R}^{n+1}\) restricted to \(\Omega \times \mathbb {R}\).