Submanifolds

Let \((M^N, g)\) be a Riemannian manifold and \(\iota: \Sigma^n \hookrightarrow M\) be a smooth immersion. The induced metric on \(\Sigma\) is defined by

so that \(\bar{g}(X,Y) = g(d\iota(X), d\iota(Y))\) for tangent vectors \(X, Y \in T\Sigma\).

The vector-valued second fundamental form \(\vec{A}\) is defined by

where \(\nabla\) is the Levi-Civita connection of \(M\) and \((\cdot)^\perp\) denotes the projection onto the normal bundle of \(\Sigma\) in \(M\). The mean curvature vector \(\vec{H}\) is defined as the trace of \(\vec{A}\):

where \(\{e_i\}\) is an orthonormal basis of \(T\Sigma\) with respect to the induced metric \(\bar{g}\).

A submanifold \(\Sigma\) is called a hypersurface when \(\mathrm{codim}(\Sigma) = 1\), i.e., \(n = N-1\).

If we assume \(\Sigma\) is a two-sided hypersurface, then there exists a globally defined unit normal vector field \(\nu\) on \(\Sigma\). In this case, the second fundamental form can be expressed as a scalar-valued symmetric \((0,2)\)-tensor \(A\) defined by

Then the mean curvature \(H=g(\vec{H}, \nu)\) is the trace of \(A\).

The eigenvalues \(\kappa_1, \ldots, \kappa_{n}\) of \(A\) are called the principal curvatures.