For a project I'm working on, I'm quite stumped on a particular Riemannian geometry calculation. Suppose I am given a $d$-dimensional Riemannian manifold with some metric given by $g_{ab}$, a hypersurface given by the zero locus of the defining function $\sigma$, and that $|\nabla \sigma|_{\Sigma} = 1$. Further, I define $n = \nabla \sigma$. How do I compute the following along the hypersurface, written in terms of intrinsic and extrinsic hypersurface curvatures?

$\nabla_n (n^a \text{Ric}_{ab} n^b)$,

$\nabla^2_n (n^a \text{Ric}_{ab} n^b)$,

$\nabla_n (\text{Ric}_{ab} \nabla^a n^b)$,

$\nabla_n R$,

$\nabla_n^2 R$, etc.

I've thought that I could use something like the theorema egregium, but that's a relationship between intrinsic curvatures and extrinsic curvatures restricted to the hypersurface, and in particular, I expect contains no information about jets of ambient extensions of the hypersurface quantities it is relating.

Is there some analogue of the theorema that contains information about jets of extensions of the present quantities to the ambient space?


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