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--- second_fundamental_form [2023/06/19 15:15] – [Mean curvature] spencer
+++ second_fundamental_form [2023/06/19 15:16] (current) – [Second Fundamental Form] spencer
@@ Line 2: / Line 2: @@
 The second fundamental form (which I usually write II and pronounce "two", but here will write $Q$) is defined for an immersed hypersurface in a Riemannian manifold; frequently the case of interest is a geodesic sphere.
-If $\nu$ denotes the (outward) unit normal to immersed hypersurface $M^{n-1} \subset (N^n, g)$ and $eta$ denotes its metric dual, then
+If $\nu$ denotes the (outward) unit normal to immersed hypersurface $M^{n-1} \subset (N^n, g)$ and $\eta$ denotes its metric dual, then
 $$Q(X,Y) := g(\nabla_X \nu, Y).$$
 Complete $\nu$ to a frame $e_1, \ldots, e_{n-1}, e_n := \nu$ around a point of $M$ with coframe $\omega^1, \ldots, \omega^n := \eta$.
-Then $\nabla_{e_i} \nu = h_{ij} \omega^j$ for some $h_{ij}$ and we accordingly define on forms that
+Then $\nabla_{e_i} \eta = h_{ij} \omega^j$ for some $h_{ij}$ and we accordingly define on forms that
 $$Q(\alpha) = h_{ij} e(\omega^i)e^*(\omega^j) \alpha.$$
 The second fundamental form is symmetric, hence diagonalizable; frequently we like to write it in this diagonal form $$Q(\alpha) = \sum_{i=1}^{n-1} \lambda_i e(\omega^i) e^*(\omega^i) \alpha.$$