Let $M$ be a smooth manifold without boundary, not necessarily compact. Let $f$ be a smooth function on $M\times M$. We say $f$ is good if for any point $(x,y)\in M\times M$ with local coordinates $x_i,y_j$ nearby, the matrix $\left[\dfrac{\partial^2f}{\partial x_i\partial y_j}\right]$ is nondegenerate. It should be clear that this concept is well-defined. I am wondering for what $M$, such a good function may exist.

Clearly, $\mathbb{R}^n$ admits such good function by setting $f(x,y)=x\cdot y$. It seems I can prove that if such a good function exists, then $M$ is orientable and affine (admitting a flat connection). It also occurs to be that flat torus $T^n$ cannot support such a function.

Thank you!

Let $M$ be a smooth manifold without boundary, not necessarily compact. Let $f$ be a real-valued smooth function on $M\times M$. We say $f$ is good if for any point $(x,y)\in M\times M$ with local coordinates $x_i,y_j$ nearby, the matrix $\left[\dfrac{\partial^2f}{\partial x_i\partial y_j}\right]$ is nondegenerate. It should be clear that this concept is well-defined. I am wondering for what $M$, such a good function may exist.

Clearly, $\mathbb{R}^n$ admits such good function by setting $f(x,y)=x\cdot y$. It seems I can prove that if such a good function exists, then $M$ is orientable and affine (admitting a flat connection). It also occurs to be that flat torus $T^n$ cannot support such a function.

Thank you!