Notes

User Tools

Site Tools

Trace:
second_fundamental_form

Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Both sides previous revisionPrevious revision
second_fundamental_form [2023/06/19 15:15] – [Second Fundamental Form] spencersecond_fundamental_form [2023/06/19 15:16] (current) – [Second Fundamental Form] spencer
Line 5: Line 5:
 $$Q(X,Y) := g(\nabla_X \nu, Y).$$ $$Q(X,Y) := g(\nabla_X \nu, Y).$$
 Complete $\nu$ to a frame $e_1, \ldots, e_{n-1}, e_n := \nu$ around a point of $M$ with coframe $\omega^1, \ldots, \omega^n := \eta$. 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
 $$Q(\alpha) = h_{ij} e(\omega^i)e^*(\omega^j) \alpha.$$ $$Q(\alpha) = h_{ij} e(\omega^i)e^*(\omega^j) \alpha.$$
 The second fundamental form is symmetric, hence diagonalizable; frequently we like to write it in this diagonal form $$Q(\alpha) = \sum_{i=1}^{n-1} \lambda_i e(\omega^i) e^*(\omega^i) \alpha.$$ The second fundamental form is symmetric, hence diagonalizable; frequently we like to write it in this diagonal form $$Q(\alpha) = \sum_{i=1}^{n-1} \lambda_i e(\omega^i) e^*(\omega^i) \alpha.$$
second_fundamental_form.1687202152.txt.gz · Last modified: by spencer