Topic
Diffeomorphism
About: Diffeomorphism is a research topic. Over the lifetime, 4085 publications have been published within this topic receiving 71134 citations.
Papers published on a yearly basis
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Papers
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1 Jan 1998TL;DR: A group theoretical approach to hydrodynamics is proposed in this article, where the authors consider the hydrodynamic geometry of diffeomorphism groups and the principle of least action implies that the motion of a fluid is described by geodesics on the group in the right-invariant Riemannian metric given by the kinetic energy.
Abstract: A group theoretical approach to hydrodynamics considers hydrodynamics to be the differential geometry of diffeomorphism groups. The principle of least action implies that the motion of a fluid is described by the geodesics on the group in the right-invariant Riemannian metric given by the kinetic energy. Investigation of the geometry and structure of such groups turns out to be useful for describing the global behavior of fluids for large time intervals.
1,728 citations
TL;DR: In this article, the Lagrangian analogue of Lie-Poisson Hamiltonian systems is defined on semidirect product Lie algebras, and an abstract Kelvin-Noether theorem for these equations is derived.
1,256 citations
TL;DR: In this article, simple algebraic relations between the algebra of functions on a manifold and its infinitesimal length elementds were shown to give the SM Lagrangian coupled to gravity.
Abstract: We first exhibit in the commutative case the simple algebraic relations between the algebra of functions on a manifold and its infinitesimal length elementds. Its unitary representations correspond to Riemannian metrics and Spin structure whileds is the Dirac propagatords=x−x=D−1, whereD is the Dirac operator. We extend these simple relations to the non-commutative case using Tomita's involutionJ. We then write a spectral action, the trace of a function of the length element, which when applied to the non-commutative geometry of the Standard Model will be shown ([CC]) to give the SM Lagrangian coupled to gravity. The internal fluctuations of the non-commutative geometry are trivial in the commutative case but yield the full bosonic sector of SM with all correct quantum numbers in this slightly non-commutative case. The group of local gauge transformations appears spontaneously as a normal subgroup of the diffeomorphism group.
1,120 citations
TL;DR: In this article, a quantization of diffeomorphism invariant theories of connections is studied and the quantum diffeomorphicism constraint is solved and the space of solutions is equipped with an inner product that is shown to satisfy the physical reality conditions.
Abstract: Quantization of diffeomorphism invariant theories of connections is studied and the quantum diffeomorphism constraint is solved. The space of solutions is equipped with an inner product that is shown to satisfy the physical reality conditions. This provides, in particular, a quantization of the Husain–Kuchař model. The main results also pave the way to quantization of other diffeomorphism invariant theories such as general relativity. In the Riemannian case (i.e., signature ++++), the approach appears to contain all the necessary ingredients already. In the Lorentzian case, it will have to be combined in an appropriate fashion with a coherent state transform to incorporate complex connections.
743 citations
TL;DR: In this paper, the existence of stable manifolds is proved almost everywhere with respect to everyf-invariant probability measure on a compact manifold M. The proof of this stable manifold theorem is through the study of random matrix products (multiplicative ergodic theorem) and perturbation of such products.
Abstract: Iff is a C1 + ɛ diffeomorphism of a compact manifold M, we prove the existence of stable manifolds, almost everywhere with respect to everyf-invariant probability measure on M. These stable manifolds are smooth but do not in general constitute a continuous family. The proof of this stable manifold theorem (and similar results) is through the study of random matrix products (multiplicative ergodic theorem) and perturbation of such products.
736 citations
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Performance Metrics
| Year | Papers |
|---|---|
| 2026 | 1 |
| 2025 | 100 |
| 2024 | 117 |
| 2023 | 314 |
| 2022 | 1,003 |
| 2021 | 192 |