# Variational Principle for Spatially Homogeneous Cosmologies/KK-theory

1. Jul 15, 2014

### center o bass

These questions applies to both spatially homogenous cosmological models, and multidimensional Kaluza-Klein theories:

Suppose we have a manifold M, of dimension m, for which there is a transitive group of isometries acting on some n-dimensional homogeneous subspace N of M. Thus there exists a basis of n killing vectors $\{\xi_1, \ldots, \xi_n\}$ on N with structure constants $C_{ij}^k$.

In this paper (http://arxiv.org/abs/gr-qc/9804043), after Equation (10.2), it is claimed that in order for Hilbert's variational principle
$$\delta \int R \text{vol}^{m}=0$$
to be compatible with the homogeneity of $N\subset M$, then the volume form $\text{vol}^n = \omega^1 \wedge \cdots \wedge \omega^n$, where $\omega^i$ are the dual basis form to $\xi_i$ ($\text{vol}^m = \text{vol}^{m-n} \wedge \text{vol}^{n}$), must be invariant with respect to the group of isometries. A requirement that is translated into $C^k_{kj} = 0$.

Additionally the author claims that $C^k_{kj} = 0$ ensures that $\mathcal{L}_{\xi_i} \delta g= 0$, where g is the metric tensor on M.

I thus wonder:
1. Why does $\text{vol}^n = \omega^1 \wedge \cdots \wedge \omega^n$ have to be invariant (with respect to the group of isometries) to make the variational Hilbert principle compatible with the homogeneity of N?
2. Why is this only true iff $C^{k}_{kj} =0$? (Is this related to the Haar measure of the group of isometries being bi-invariant?)
3. How does $C^k_{kj} = 0$ ensure $\mathcal{L}_{\xi_i} \delta g= 0$ ?

2. Jul 30, 2014