Journal•ISSN: 0926-2245
Differential Geometry and Its Applications
Elsevier BV
About: Differential Geometry and Its Applications is an academic journal published by Elsevier BV. The journal publishes majorly in the area(s): Curvature & Scalar curvature. It has an ISSN identifier of 0926-2245. Over the lifetime, 1748 publications have been published receiving 21052 citations.
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TL;DR: In this article, it was shown that there are no compact three-dimensional Ricci solitons other than spaces of constant curvature, which generalizes a result obtained for surfaces by Hamilton.
Abstract: In this short article we show that there are no compact three-dimensional Ricci solitons other than spaces of constant curvature. This generalizes a result obtained for surfaces by Hamilton [4]. The proof involves a careful analysis of the ODE for the curvature which is associated to the Ricci flow.
343 citations
TL;DR: In this article, the geometrical foundations of first order Lagrangian and Hamiltonian field theories are clarified by introducing in a systematic way multisymplectic manifolds, the field theoretical analogues of the symplectic structures used in Geometrical mechanics.
Abstract: The general purpose of this paper is to attempt to clarify the geometrical foundations of first order Lagrangian and Hamiltonian field theories by introducing in a systematic way multisymplectic manifolds, the field theoretical analogues of the symplectic structures used in geometrical mechanics. Much of the confusion surrounding such terms as gauge transformation and symmetry transformation as they are used in the context of Lagrangian theory is thereby eliminated, as we show. We discuss Noether's theorem for general symmetries of Lagrangian and Hamiltonian field theories. The cohomology associated to a group of symmetries of Hamiltonian or Lagrangian field theories is constructed and its relation with the structure of the current algebra is made apparent.
250 citations
TL;DR: In this article, a metric quasi-Einstein metric is defined, where the m -Bakry-Emery Ricci tensor is a constant multiple of the metric tensor.
Abstract: We call a metric quasi-Einstein if the m -Bakry–Emery Ricci tensor is a constant multiple of the metric tensor. This is a generalization of Einstein metrics, it contains gradient Ricci solitons and is also closely related to the construction of the warped product Einstein metrics. We study properties of quasi-Einstein metrics and prove several rigidity results. We also give a splitting theorem for some Kahler quasi-Einstein metrics.
209 citations
TL;DR: In this paper, the authors give a topological and geometrical description of focus-focus singularities of integrable Hamiltonian systems and explain why the monodromy around these singularities is non-trivial.
Abstract: We give a topological and geometrical description of focus-focus singularities of integrable Hamiltonian systems. In particular, we explain why the monodromy around these singularities is non-trivial, a result obtained before by J.J. Duistermaat and others for some concrete systems.
145 citations
TL;DR: In this paper, the authors derive a decomposition theorem for an isometric immersion f : M → N κ n of a warped product M into the standard n -space of constant curvature κ and classify all warped product decompositions.
Abstract: We derive a decomposition theorem for an isometric immersion f : M → N κ n of a warped product M into the standard n -space N κ n of constant curvature κ and classify all warped product decompositions of the standard spaces.
135 citations