Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Consider $\Sigma^n\subset {\bf R}^{n+p}$ a submanifold and $\{e_i, e_\alpha\}$ an orthonormal frame where the Greek indexes stand for the normal vectors. Then define

$H^{\alpha}_{ij,k}=e_k\langle e_{\alpha},\nabla _{e_i} e_j\rangle $

Can anyone provide a proof that $H^{\alpha}_{ij,k}=H^{\alpha}_{ik,j}$ for submanifolds in ${\bf R}^n$?

PS: The $e_k$ vector is deriving the function.

share|cite|improve this question
I would tentatively suggest that $\nabla_{e_i} e^j=\delta^j_i$. And so therefore $H^\alpha_{ij,k}=e_k e_a\delta^j_i$ – Keith Afas Feb 22 at 1:02

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.