Let $\{h_i(x)\}_{i=1}^k$ be a set of polynommials on $\mathbb{R}^m$ and $m > k$. Let $M$ be the set defined the polynomilas , i.e. $$ M=\{x \in \mathbb{R}^m \, | \, \wedge_i h_i(x)=0\} $$ Assume that $M$ is path=connected and every $\frac{\partial h_i}{\partial x_j}$ are linear independent function. Hence $det(\frac{\partial h_i}{\partial x_j})$ can be zero at most on a null subset of $M$.

From the Implicit Function Theorem if $det(\frac{\partial h_i}{\partial x_j})\neq 0$ then $M$ is a smooth manifold and hence a Riemannian manifold. However $det(\frac{\partial h_i}{\partial x_j})=0$ for a null subset of $M$.

My question is:

Is it possible to do Riemannian geometry on $M$?

In particular I am interested to use Hopf-Rinow Theorem on $M$?

Thanks in advance.

  • 1
    $\begingroup$ An example: $M=\{ (x, y)\in \mathbb R^2\ :\ xy=0\}$. This is the set consisting of the two coordinate axes. Can you do Riemannian geometry on it? Yes but only away from the origin. $\endgroup$ – Giuseppe Negro Nov 15 '17 at 20:55
  • $\begingroup$ Yes, but my interest is to use Hopf-Rinow Theorem on $M$. Is it possible ? $\endgroup$ – jaogye Nov 16 '17 at 5:36

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