# Generalization of principal, Gaussian and mean curvatures to surfaces in higher-dimensional manifolds

Let $$M$$ be an $$n$$-dimensional Riemannian manifold and $$\Sigma\subset M$$ an embedded submanifold of dimension two, i.e., a surface. Its second-fundamental form $${\text{II}}:\Gamma(T\Sigma)\times \Gamma(T\Sigma)\to \Gamma(N\Sigma)$$ takes values in sections of the normal bundle. If $$\{N_i\}\subset \Gamma(N\Sigma)$$ is an orthonormal basis of sections of the normal bundle, we can decompose the second fundamental form as

$$\text{II}(X,Y) = \sum_{i=1}^{n-2} \text{II}^i(X,Y) N_i,$$

in which $$\text{II}^i:\Gamma(T\Sigma)\times\Gamma(T\Sigma)\to C^\infty(\Sigma)$$. We can then associate a $$2\times 2$$ matrix of components for each $$\text{II}^i$$ and therefore we have a collection of $$(n-2)$$ such matrices.

In $$n=3$$ there is just a single matrix. Its eigenvalues are defined as the principal curvatures $$\kappa_1$$ and $$\kappa_2$$, and from them we can construct the Gaussian curvature $$G=\kappa_1\kappa_2$$ and the mean curvature $$H = \frac{\kappa_1+\kappa_2}{2}$$.

What happens in this more general case? What is the appropriate generalization of the principal, Gaussian and mean curvatures to $$n>3$$? What is the geometric data contained in the second fundamental form analogous to these quantities $$\kappa_1, \kappa_2, G, H$$ in $$n=3$$?

The obvious thing would be to consider for each of the $$\text{II}^i$$ an associated pair of $$\kappa_1^i$$ and $$\kappa_2^i$$ with its associated $$G^i$$ and $$H^i$$, but this seems quite random and doesn't seem to be the right quantities to consider.

• It sounds like $\sigma_k$ curvatures, you take the eigenvalues or principle curvatures and make symmetric $k^{th}$-degree polys. The extreme cases being the trace or determinant which are respectively Gaussian and (up to averaging constant) Mean curvatures. I'll have to reread to see if this is what you are looking for, but it is for higher dimensional matrices. Commented Jan 24 at 20:24
• That sounds interesting. Do you have a reference on that?
– Gold
Commented Jan 24 at 20:30
• Try John Lee's "Riemannian Manifolds: An Introduction to Curvature", Ch.8 Commented Jan 24 at 20:37
• I think it was something the professor mentioned actually, maybe not explicitly mentioned in the text... Commented Jan 24 at 20:41
• The net Gaussian curvature (defined for any surface with a Riemannian metric) is the sum of your $n-2$ Gaussian curvatures. There is no analogue of mean curvature. You might also be interested in reading about Lipschitz-Killing curvature. Commented Jan 25 at 6:18

First, let's review what happens when $$n=3$$. In that case, there is a unique (up to sign) normal vector $$N$$ (also known as the Gauss map), and at each point $$p \in \Sigma$$, the second fundamental form is a symmetric bilinear tensor $$\operatorname{II}: T_p\times T_p \rightarrow \mathbb{R},$$ where for any $$v, w \in T_p\Sigma$$, $$\operatorname{II}(v,w) = g_M(v,\nabla_wN) = -g_M(\nabla_vw,N),$$ where $$\nabla$$ denotes the Levi-Civita connection of $$g_M$$ (and not $$g_\Sigma$$). If $$(e_1,e_2)$$ is an orthonormal basis of $$T_pM$$, then the mean curvature is defined to be $$H(p) = \operatorname{II}(e_1,e_1) + \operatorname{II}(e_2,e_2)$$ and the Gauss curvature is $$K(p) = \operatorname{II}(e_1,e_1)\operatorname{II}(e_2,e_2) - (\operatorname{II}(e_1,e_2))^2.$$
When $$n \ge 3$$, then at each $$p \in \Sigma$$, the second fundamental form is the bilinear map $$\operatorname{II}: T_p\times T_p \rightarrow N_p,$$ where $$N_p\subset T_pM$$ is the subspace of vectors normal to $$T_pM$$ given by $$\operatorname{II}(v,w) = \pi^\perp(\nabla_v w),$$ where $$\pi^\perp: T_pM \rightarrow N_p$$ is orthogonal projection. Then the formulas above, with scalar multiplication replaced by the inner product defined by $$g_M$$, still hold: \begin{align*} H(p) &=\operatorname{II}(e_1,e_1) + \operatorname{II}(e_2,e_2)\\ K(p) &= g_M(\operatorname{II}(e_1,e_1),\operatorname{II}(e_2,e_2)) - g_M(\operatorname{II}(e_1,e_2),\operatorname{II}(e_1,e_2)). \end{align*} $$K$$ is still the Gauss curvature and a scalar function intrinsic to $$\Sigma$$. However, $$H$$ is no longer a scalar function. It is a section of the normal bundle, i.e., $$H(p) \in N_p$$. It also is still what appears in the variational formula for the area of $$\Sigma$$.
ADDENDUM: If $$n > 3$$, the principal curvatures are also no longer well defined as scalars. However, for each unit normal $$\nu \in N_p$$, you can define the principal curvatures for that unit direction to be the eigenvalues of $$\nu\cdot\operatorname{II}$$.