\chapter{First fundamental form}


\[I_p : T_p \Sigma t\o \mathbb{R}\]
\[I_p (\mathbf{v}) = \| \mathbf{v} \|^2\]

\chapter{Metric tensors}

Given a manifold $M$, a \emph{metric tensor on $M$ at $p$} is a nondegenerate, symmetric, bilinear form $g_p : T_p M \times T_p M \to \mathbb{R}$.