The Group of Isometries of a Riemannian Manifold

Myers, S. B.; Steenrod, N. E.

doi:10.2307/1968928

Cited by 312 publications

(193 citation statements)

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“…3 Let I(Σ, h) := {φ ∈ Diff(Σ) | φ * h = h} be the isometry group of (Σ, h), then it is well known that dim I(Σ, h) ≤ 1 2 n(n +1), where n = dim Σ. I(Σ, h) is compact if Σ is compact (see, e.g., Sect. 5 of [62]). Conversely, if Σ allows for an effective action of a compact group G then it clearly allows for a metric h on which G acts as isometries (just average any Riemannian metric over G.) The degree of symmetry of Σ, denoted by deg(Σ), is defined by deg(Σ) := sup h∈Riem(Σ) {dim I(Σ, h)}.…”

Section: Defining Superspacementioning

confidence: 97%

The Superspace of geometrodynamics

Giulini

2009

Gen Relativ Gravit

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Section: Defining Superspacementioning

confidence: 97%

The Superspace of geometrodynamics

Giulini

2009

Gen Relativ Gravit

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“…For example, Myers and Steenrod proved this fact for Riemannian Manifolds in [26], Fukaya and Yamaguchi for Alexandrov spaces with curvature bounded by above and by Gerardo Sosa gsosa@mis.mpg.de 1 Max Planck Institute for Mathematics in the Sciences, Leipzig 04103, Germany below in [12,36], and Cheeger, Colding, and Naber in the case of Ricci Limit spaces in [4,6]. In contrast, there exist metric spaces for which ISO(X) is not a Lie group, see for instance Examples 5.1 and 5.2.…”

Section: Introductionmentioning

confidence: 94%

The Isometry Group of an RCD ∗ Space is Lie

Sosa

2017

Potential Anal

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We give necessary and sufficient conditions that show that both the group of isometries and the group of measure-preserving isometries are Lie groups for a large class of metric measure spaces. In addition we study, among other examples, whether spaces having a generalized lower Ricci curvature bound fulfill these requirements. The conditions are satisfied by RCD * -spaces and, under extra assumptions, by CD-spaces, CD * -spaces, and MCP-spaces. However, we show that the MCP-condition by itself is not enough to guarantee a smooth behavior of these automorphism groups.

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“…This generalizes the theorem of H. Cartan for bounded domains. It has been proved by Meyers and Steenrod [23] that the group of isometries of a Riemannian space is a Lie group and the isotropy group at every point of M is compact. Their proof can be nowadays simplified by the theory of connections.…”

Section: Shoshichi Kobayashimentioning

confidence: 99%

Geometry of bounded domains

Kobayashi¹

1959

Trans. Amer. Math. Soc.

201

119

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In this paper, we shall study differential geometric properties of bounded domains in Cn. Here is the summary of our results.We consider an w-dimensional complex manifold M and the Hilbert space of square integrable holomorphic re-forms on M. After Bergman [3; 4; 5], we define the kernel form on M (instead of the kernel function) and, under certain assumptions, we define the invariant metric of Bergman. This method of generalizing the theory of S. Bergman (although the generalization is not essential) allows us to define the Bergman metric on certain compact complex manifolds.Some elementary properties of the kernel form and the Bergman metric (mostly already classical) are studied for the sake of completeness. Then we reexamine the theorem of H. Cartan on the group of holomorphic transformations of a bounded domain from the differential geometric point of view.We study also differential geometric properties of a manifold M which admits a discontinuous group D of holomorphic transformations such that M/D is compact. It should be noted that such a manifold possesses properties similar to those of a homogeneous manifold.Bremermann has studied the bounded domains with the following property (P): the kernel function goes to infinity at every boundary point [8]. He has shown that a bounded domain with the property (P) is a domain of holomorphy and that the converse is not true. (The same result has been obtained also by Sommer and Mehring [22].) Making use of this result, he has proved that if a bounded domain is complete with respect to the Bergman metric, then it is a domain of holomorphy.Since the kernel function is not intrinsically defined, the property (P) is not intrinsic. We consider, therefore, a condition which is stronger than (P) but which is intrinsic.

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The Group of Isometries of a Riemannian Manifold

Cited by 312 publications

References 0 publications

The Superspace of geometrodynamics

The Superspace of geometrodynamics

The Isometry Group of an RCD ∗ Space is Lie

Geometry of bounded domains

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