# How to translate this equation into physicist's notation? [closed]

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.

## closed as off-topic by Aaron Stevens, Kenshin, StephenG, ZeroTheHero, SuperCiociaSep 22 at 7:10

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