Let $(M^{n+1}, g)$ be a Lorentzian manifold (spacetime) that contains a Riemannian/spacelike hypersurface $(\Sigma ^{n},h).$ Then we can define the second fundamental form of the hypersurface in many equivalent ways, for example, decomposing $M-$ covariant derivative into tangential and normal components. This normal component gives the notion of the second fundamental form of $\Sigma$. This second fundamental form helps us write Gauss, Codazzi, and Mainardi equations nicely, eventually producing Einstein Constraint Equations.

Note that in a 1984 paper titled Cauchy formulation for Bach equations, R. Schimming defined the third and fourth fundamental forms as the following:

$III_{ij} = n^{c}n^{d} \delta^{a}_{i}\delta^{b}_{j} W_{cabd}$ and

$IV_{ij} = n^{c} \delta^{a}_{i}\delta^{b}_{j} \nabla ^{d} W_{cabd}, $ where $W$ is the Weyl curvature tensor and $n$ is a unit normal vector to the hypersurface. Note that $\delta^{a}_{i}$ is the partial derivative of the $a$th coordinate function in the $i$th direction, and indices $a, b, c,d$ are spacetime indices and $i, j$ are spatial indices.

I can see that using these third and fourth fundamental forms, Bach equations can be expressed as an initial value problem, and that's huge! However, these two definitions seem unnatural and unmotivated to me. Is there any geometric way to define these two fundamental forms like the usual first & second fundamental forms?

On Wikipedia, I see that the second fundamental form of a parametric surface in $\mathbb{R}^3$ is deduced from the second-order Tylor expansion of the defining functions of the surface. Maybe considering the fourth Taylor expansion of the function that defines the hypersurface $\Sigma$, we might define the third and fourth fundamental forms(I am not sure though). Even if we can do that, assigning some sorted geometric information into those forms would be a difficult task(maybe).

I would highly appreciate it if you could help me understand the third and fourth fundamental forms geometrically. I am sorry for my wordy problem statement. Thanks so much.


1 Answer 1


It is very unhelpful that you give the wrong title of the paper you are asking for. In reality, the paper appears to be

Cauchy's problem for Bach's equations of general relativity. R. Schimming

Banach Center Publications (1984)

Volume: 12, Issue: 1, page 225-231

A MathSciNet search shows that there is one indexed paper citing it, namely

Asymptotically Simple Solutions of the Vacuum Einstein Equations in Even Dimensions

Michael T. Anderson & Piotr T. Chruściel

Communications in Mathematical Physics volume 260, pages 557–577 (2005).

A quick look at that paper suggests that it discusses a very closely related but more general story in quite much more invariant way. Perhaps looking at it carefully will help you with your question?

  • $\begingroup$ Thanks so much. I will take a look at the paper you've suggested. $\endgroup$
    – Boka Peer
    Mar 31 at 22:29

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