Context

Riemannian geometry

Riemannian geometry

Basic definitions

Riemannian manifold

• metric, isometry, isometry group

moduli space of Riemannian metrics

pseudo-Riemannian manifold

• Lorentzian manifold, spacetime

geodesic

geodesic convexity

geodesic flow

Levi-Civita connection

• Riemann curvature, torsion of a metric connection

Further concepts

Hodge inner product, Hodge star operator

gradient, gradient flow

Theorems

• Poincaré conjecture-theorem

Applications

gravity

• Einstein-Hilbert action, Einstein equations, supergravity

Contents

• Idea

• References

• General

• As symmetries of brane dynamics

• As symmetries of unimodular gravity

• As symmetries of fractional quantum Hall systems

Idea

A diffeomorphism ϕ:X⟶X\phi \colon X \longrightarrow X of a (pseudo-) Riemannian manifold XX is volume preserving if its pullback of differential forms fixes the volume form: ϕ *vol=vol\phi^\ast vol = vol.

If the dimensio is dim(X)=2dim(X) = 2, then one also speaks of area-preserving diffeomorphisms.

Volume-preserving diffeomorphisms appear:

as the gauge symmetries of unimodular gravity?

as residual gauge symmetries of relativistic brane sigma models after light cone gauge-fixing (cf. below)

as symmetries of effective descriptions of fractional quantum Hall systems (cf. below)

References

General

• Dusa McDuff: On the Group of Volume-Preserving Diffeomorphisms of R n\mathbf{R}^n, Transactions of the AMS 261 1 (1980) [doi:10.1090/S0002-9947-1980-0576866-3]

As symmetries of brane dynamics

On volume-preserving diffeomoprhisms as residual gauge symmetries of brane sigma-models in light cone gauge:

• Eric Bergshoeff, Ergin Sezgin, Y. Tanii, Paul K. Townsend: Super pp-branes as gauge theories of volume preserving diffeomorphisms, Annals of Physics 199 2 (1990) 340-365 [doi:10.1016/0003-4916(90)90381-W]

Discussion specifically for the M2-brane sigma-model (cf. M2-brane – Light-cone quantization to the BFSS matrix model):

Emmanuel G. Floratos?, John Iliopoulos: A note on the classical symmetries of the closed bosonic membranes, Physics Letters B 201 2 (1988) 237-240 [doi:10.1016/0370-2693(88)90220-1]

Bernard de Wit, Jens Hoppe, Hermann Nicolai, On the Quantum Mechanics of Supermembranes, Nucl. Phys. B 305 (1988) 545-581 [doi:10.1016/0550-3213(88)90116-2, spire:261702, pdf, pdf]

See also:

• Y. Matsuo, Y. Shibusa: Volume Preserving Diffeomorphism and Noncommutative Branes, JHEP 0102:006 (2001) [arXiv:hep-th/0010040]

As symmetries of unimodular gravity

(…)

As symmetries of fractional quantum Hall systems

As symmetries of fractional quantum Hall systems (cf. supersymmetry in FQH systems):

• Yi-Hsien Du: Chiral Graviton Theory of Fractional Quantum Hall States [arXiv:2509.04408]

Created on January 8, 2026 at 16:37:38. See the history of this page for a list of all contributions to it.