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.