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Get the classical Gauss's equation $\langle R_{x,y}Z,W \rangle = \langle X,W \rangle \langle Y,Z \rangle - \langle X,Z \rangle \langle Y,W \rangle - \langle A(x), Z \rangle \langle A(y), W \rangle + \langle A(x), W \rangle \langle A(y),Z \rangle $

$X,Y,Z,W \in TM$

Since $\phi:M^n \implies S^{n+1}$ is minimal immersion and A the second fundamental form

I want calculate the trace twice to show $K=1-\frac{1}{n(n-1)}||A||^2$, K is the scalar curvature.

any ideia?

Thank you advance!

Fabricio Gonçalves
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  • Have you learned what the trace of a tensor is? – Deane Jul 31 '23 at 18:05
  • I thought as follows:

    $Ric(x,y) = \sum_{k=1}^{n} \langle R_{x,e_k}Y, e_k \rangle = \sum_{k=1}^{n} { \langle x,e_k \rangle \langle y,e_k \rangle - \langle x,y \rangle \langle e_k, e_k \rangle } - \sum_{k=1}^{n} { \langle A(x), e_k \rangle \langle A(e_k),y \rangle - \langle A(x), y \rangle \langle A(e_k),e_k \rangle } $

    $x,y \in TM$ and ${ e_1,....,e_k }$ is a orthonormal basis. But I can't conclude

     – Fabricio Gonçalves Jul 31 '23 at 18:36

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