induced Riemannian metric

Pullback of Riemannian metric

Let $f:M\to N$ be an immersion between a smooth manifold $M$ and a Riemannian manifold $(N,g)$. Then there exists an induced metric $f^{\ast }g$ using the pullback of Riemannian metric on $M$

$$(f^g{)}_p(X,Y)=g_{f(p)}(d{f}_p(X_p),d{f}_p(Y_p))$$

NOTE

We need $f$ to be an immersion, so for $X\ne 0$ then $f^{\ast }X\ne 0$. This makes sure that $f^{\ast }g$ is still positive definite.

Examples

$(S^2,g)$ - with spherical coordinates

todo

$S^2$ with stereographic coordinates

todo