Curvature of multivariate polynomial at the origin

Question

For any Gaussian curvature $K$, the polynomial $x^2 + \frac{K}{2}y^2$ has Gaussian curvature of K at the origin. Let $M$ be a Riemann Manifold of dimension $n$ and let $x$ be a point in $M$. Does there exist a polynomial $P$ such that the local geometry at the origin of $P$ is isometric to the local geometry of $x$?