# Leaf of Leaf Foliation and Beltrami Parametrization in $d>2$ dimensional Gravity

Abstract : This work shows the existence of a $d>2$ dimensional covariant "Beltrami vielbein" that generalizes the $d=2$ situation. Its definition relies on a covariant sub-foliation of the Arnowitt Deser Misner leafs of $d$-dimensional Lorentzian manifolds ${\cal M}_d$, $\Sigma^{ADM}_{d-1}= \Sigma_{d-3} \times \Sigma_2$. $\Sigma_2$ is the sub-foliating randomly varying Riemann surface. The "Beltrami d-bein" is parametrized by ${d(d+1)}/2$ independent fields belonging to different categories, each one with a specific interpretation. The Weyl invariant sector beautifully selects the $d(d-3)/2$ physical local degrees of freedom of $d$-dimensional gravity. Given a generic $d$-bein with its $d^2$ independent field components, the construction of the corresponding Beltrami d-bein is made possible by a covariant gauge fixing of the Lorentz gauge symmetry in the tangent space over each point of ${\cal M}_d$. There is thus a one to one correspondance between the components of the Beltrami d-bein and those of the associated Beltrami $d$-metric, the latter being quadratic functions of the former. The computation of the Spin connection and of the Einstein action in function of the Beltrami fields delivers interesting expressions. A gravitational "physical gauge" choice is introduced that takes advantage of the geometrical specificities of the Beltrami parametrization of gravitational field variables. Further restrictions may simplify the $d$-dimensional Beltrami parametrization when ${\cal M}_d$ has a given spatial holonomy. The latter point is exemplified in the case of $d=8$ spaces with $G_2\subset SO(1,7)$ holonomy. The Lorentzian results presented in this paper can be extended to the Euclidean case.
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https://hal.archives-ouvertes.fr/hal-03356521
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Submitted on : Tuesday, September 28, 2021 - 10:35:22 AM
Last modification on : Thursday, September 30, 2021 - 3:37:54 AM

### Citation

Laurent Baulieu. Leaf of Leaf Foliation and Beltrami Parametrization in $d>2$ dimensional Gravity. 2021. ⟨hal-03356521⟩

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