scholarly journals Reduction Theorem for Connections and its Application to the Problem of Isotropy and Holonomy Groups of a Riemannian Manifold

1955 ◽  
Vol 9 ◽  
pp. 57-66 ◽  
Author(s):  
Katsumi Nomizu
Keyword(s):  
Differential Geometry ◽  
Riemannian Manifold ◽  
Symmetric Space ◽  
Homogeneous Spaces ◽  
Holonomy Group ◽  
Complete Riemannian Manifold ◽  
Reduction Theorem ◽  
Riemannian Symmetric Space ◽  
Isotropy Group ◽  
Holonomy Groups

The present paper constitutes, together with [13], a continuation of the study of differential geometry of homogeneous spaces which we started in [11]. Our main result states that if the homogeneous holonomy group of a complete Riemannian manifold is contained in the linear isotropy group at each point, then the Riemannian manifold is symmetric. The converse is of course one of the well known properties of a Riemannian symmetric space [4]. Besides the results already sketched in [12], we add a few applications of the main theorem.

scholarly journals On Infinitesimal Holonomy and Isotropy Groups

1957 ◽  
Vol 11 ◽  
pp. 111-114 ◽  
Author(s):  
Katsumi Nomizu
Keyword(s):  
Riemannian Manifold ◽  
Curvature Tensor ◽  
Tensor Field ◽  
Present Note ◽  
General Theorem ◽  
Holonomy Group ◽  
Complete Riemannian Manifold ◽  
Isotropy Group ◽  
Covariant Derivatives ◽  
Derivatives Of

We have proved in [2] that if the restricted homogeneous holonomy group of a complete Riemannian manifold is contained in the linear isotropy group at every point, then the Riemannian manifold is locally symmetric, that is, the covariant derivatives of the curvature tensor field are zero. The proof of this theorem, however, depended on an insufficiently stated proposition (Theorem 1, [2]). In the present note, we shall give a proof of a more general theorem of the same type.

scholarly journals A Generalized Bochner Technique and Its Application to the Study of Conformal Mappings

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 333
Author(s):  
Vladimir Rovenski ◽  
Sergey Stepanov ◽  
Irina Tsyganok
Keyword(s):  
Differential Geometry ◽  
Riemannian Manifold ◽  
Convex Functions ◽  
Complete Riemannian Manifold ◽  
Conformal Mappings ◽  
Global Behavior ◽  
Bochner Technique ◽  
Kählerian Manifolds

This article is devoted to geometrical aspects of conformal mappings of complete Riemannian and Kählerian manifolds and uses the Bochner technique, one of the oldest and most important techniques in modern differential geometry. A feature of this article is that the results presented here are easily obtained using a generalized version of the Bochner technique due to theorems on the connection between the geometry of a complete Riemannian manifold and the global behavior of its subharmonic, superharmonic, and convex functions.

scholarly journals Infinitesimal Holonomy Groups of Bundle Connections

1956 ◽  
Vol 10 ◽  
pp. 105-123 ◽  
Author(s):  
Hideki Ozeki
Keyword(s):  
Differential Geometry ◽  
Present Note ◽  
Holonomy Group ◽  
Fiber Bundles ◽  
Linear Connections ◽  
Holonomy Groups ◽  
Principal Fiber Bundles ◽  
Sharpened Form

In Introduction In differential geometry of linear connections, A. Nijenhuis has introduced the concepts of local holonomy group and infinitesimal holonomy group and obtained many interesting results [6].The purpose of the present note is to generalize his results to the case of connections in arbitrary principal fiber bundles with Lie structure groups. The concept of local holonomy group can be immediately generalized and has been already utilized by S. Kobayashi [4]. Our main results are Theorems 4 and 5 on infinitesimal holonomy groups. The proofs depend on a little sharpened form of a theorem of Ambrose-Singer [1]. In the case of linear connections, our infinitesimal holonomy group coincides with that of Nijenhuis, as we shall show in Section 6.

CRITERION OF PROPER ACTIONS ON HOMOGENEOUS SPACES OF CARTAN MOTION GROUPS

2007 ◽  
Vol 18 (03) ◽  
pp. 245-254
Author(s):  
TARO YOSHINO
Keyword(s):  
Symmetric Space ◽  
Lie Group ◽  
Homogeneous Spaces ◽  
Locally Compact Group ◽  
Riemannian Symmetric Space ◽  
Locally Compact ◽  
Motion Group ◽  
Discrete Subgroups ◽  
Proper Actions ◽  
Motion Groups

The Cartan motion group associated to a Riemannian symmetric space X is a semidirect product group acting isometrically on its tangent space. For two subsets in a locally compact group G, Kobayashi introduced the concept of "properness" as a generalization of properly discontinuous actions of discrete subgroups on homogeneous spaces of G. In this paper, we give a criterion of properness for homogeneous spaces of Cartan motion groups. Our criterion has a similar feature to the case where G is a reductive Lie group.

scholarly journals THE CANONICAL EIGHT-FORM ON MANIFOLDS WITH HOLONOMY GROUP SPIN(9)

2010 ◽  
Vol 07 (07) ◽  
pp. 1159-1183 ◽  
Author(s):  
M. CASTRILLÓN LÓPEZ ◽  
P. M. GADEA ◽  
I. V. MYKYTYUK
Keyword(s):  
Riemannian Manifold ◽  
Explicit Expression ◽  
Curvature Tensor ◽  
Holonomy Group ◽  
Canonical Forms ◽  
Group Spin ◽  
Holonomy Groups ◽  
Explicit Expressions

An explicit expression of the canonical 8-form on a Riemannian manifold with a Spin(9)-structure, in terms of the nine local symmetric involutions involved, is given. The list of explicit expressions of all the canonical forms related to Berger's list of holonomy groups is thus completed. Moreover, some results on Spin(9)-structures as G-structures defined by a tensor and on the curvature tensor of the Cayley planes, are obtained.

scholarly journals On weak holonomy

2005 ◽  
Vol 96 (2) ◽  
pp. 169 ◽  
Author(s):  
Bogdan Alexandrov
Keyword(s):  
Riemannian Manifold ◽  
Lie Groups ◽  
Holonomy Group ◽  
Holonomy Groups

We prove that $\mathrm{SU}(n)$ ($n \ge 3$) and $\mathrm{Sp}(n)U(1)$ ($n \ge 2$) are the only connected Lie groups acting transitively and effectively on some sphere which can be weak holonomy groups of a Riemannian manifold without having to contain its holonomy group. In both cases the manifold is Kähler.

HARNACK-TYPE INEQUALITIES FOR THE POROUS MEDIUM EQUATION ON A MANIFOLD WITH NON-NEGATIVE RICCI CURVATURE

2012 ◽  
Vol 23 (04) ◽  
pp. 1250009 ◽  
Author(s):  
JEONGWOOK CHANG ◽  
JINHO LEE
Keyword(s):  
Porous Medium ◽  
Riemannian Manifold ◽  
Ricci Curvature ◽  
Porous Medium Equation ◽  
Complete Riemannian Manifold ◽  
Gradient Estimates ◽  
Medium Equation

We derive Harnack-type inequalities for non-negative solutions of the porous medium equation on a complete Riemannian manifold with non-negative Ricci curvature. Along with gradient estimates, reparametrization of a geodesic and time rescaling of a solution are key tools to get the results.

scholarly journals Some results on stable p-harmonic maps

1994 ◽  
Vol 36 (1) ◽  
pp. 77-80 ◽  
Author(s):  
Leung-Fu Cheung ◽  
Pui-Fai Leung
Keyword(s):  
Riemannian Manifold ◽  
Critical Point ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Energy Functional ◽  
Complete Riemannian Manifold ◽  
Image Position ◽  
Simply Connected

For each p ∈ [2, ∞)a p-harmonic map f:Mm→Nn is a critical point of the p-energy functionalwhere Mm is a compact and Nn a complete Riemannian manifold of dimensions m and n respectively. In a recent paper [3], Takeuchi has proved that for a certain class of simply-connected δ-pinched Nn and certain type of hypersurface Nn in ℝn+1, the only stable p-harmonic maps for any compact Mm are the constant maps. Our purpose in this note is to establish the following theorem which complements Takeuchi's results.

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