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I asked this in math stackexchange but no one has answered there so I ask here.

How to translate this equation into physicist's notation, i.e. tensors with indices? $$\left\langle R_{N}\left(u,v\right)w,z\right\rangle =\left\langle R_{M}\left(u,v\right)w,z\right\rangle +\left\langle \mathbb{I}\left(u,z\right),\mathbb{I}\left(v,w\right)\right\rangle -\left\langle \mathbb{I}\left(u,w\right),\mathbb{I}\left(v,z\right)\right\rangle $$ This equation is from the Wikipedia article about second fundamental form, last equation of that article. $R_N$ is curvature tensor of a manifold embedded in another manifold with curvature tensor $R_M$. $\mathbb{I}$ is the second fundamental form.

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Well, $R$ is a tensor which eats three vectors ($u,v,$ and $w$) and produces a vector, so it is a $(1,3)-$tensor. Similarly, $\mathbb I$ eats two vectors and produces a vector, so it is a $(1,2)-$tensor.

Expanding in some basis $\{\hat e_i\}$,

$$R(u,v)w = u^jv^k w^l R^i_{\ jkl} \hat e_i$$ and $$\mathbb I(u,v)= u^jv^k \mathbb I^i_{\ jk} \hat e_i$$

From there, the bilinearity of the inner product allows you to write e.g.

$$\langle R(u,v)w,z\rangle = u^jv^kw^lz^m R^i_{\ jkl} \langle \hat e_i,\hat e_m\rangle$$ $$\equiv u^jv^kw^lz^m R^i_{\ jkl} g_{im}$$

where $g_{im}$ are the components of the metric tensor. You can follow precisely the same procedure on each of the other three terms and then factor out the vector components $u^jv^kw^lz^m$ to get a relationship between the components of $R_N,R_M$, and $\mathbb I$.

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