scholarly journals Almost All Finsler Metrics have Infinite Dimensional Holonomy Group

2020 ◽  
Author(s):  
B. Hubicska ◽  
V. S. Matveev ◽  
Z. Muzsnay
Keyword(s):  
Dense Subset ◽  
Holonomy Group ◽  
Finsler Metrics ◽  
Infinite Dimensional ◽  
Holonomy Groups ◽  
Almost All

Abstract We show that the set of Finsler metrics on a manifold contains an open everywhere dense subset of Finsler metrics with infinite-dimensional holonomy groups.

The Set of Finite Operators is Nowhere Dense

1989 ◽  
Vol 32 (3) ◽  
pp. 320-326 ◽  
Author(s):  
Domingo A. Herrero
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Dense Subset ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Dimensional Hilbert Space

AbstractA bounded linear operator A on a complex, separable, infinite dimensional Hilbert space is called finite if for each . It is shown that the class of all finite operators is a closed nowhere dense subset of

scholarly journals On the Holonomy Groups of Linear Connections

1956 ◽  
Vol 10 ◽  
pp. 97-100 ◽  
Author(s):  
Jun-Ichi Hano ◽  
Hideki Ozeki
Keyword(s):  
Linear Group ◽  
Affine Space ◽  
General Linear Group ◽  
Affine Connection ◽  
Holonomy Group ◽  
Linear Connection ◽  
General Linear ◽  
Linear Connections ◽  
Holonomy Groups ◽  
Lie Subgroup

In this note we show in § 1, as the main result, that any connected Lie subgroup of the general linear group GL(n, R) can be realized as the holonomy group of a linear connection, i.e. the homogeneous holonomy group of the associeted affine connection, defined on an affine space of dimension n (n ≧ 2).

scholarly journals Projectively flat Finsler manifolds with infinite dimensional holonomy

2015 ◽  
Vol 27 (2) ◽  
Author(s):  
Zoltán Muzsnay ◽  
Péter T. Nagy
Keyword(s):  
Lie Group ◽  
Holonomy Group ◽  
Finsler Manifold ◽  
Flag Curvature ◽  
Compact Lie Group ◽  
Finsler Manifolds ◽  
Infinite Dimensional ◽  
Constant Flag Curvature ◽  
Projectively Flat

AbstractRecently, we developed a method for the study of holonomy properties of non-Riemannian Finsler manifolds and obtained that the holonomy group cannot be a compact Lie group if the Finsler manifold of dimension >2 has non-zero constant flag curvature. The purpose of this paper is to move further, exploring the holonomy properties of projectively flat Finsler manifolds of non-zero constant flag curvature. We prove in particular that projectively flat Randers and Bryant–Shen manifolds of non-zero constant flag curvature have infinite dimensional holonomy group.

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 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.

scholarly journals A Note on Infinitesimal Holonomy Groups

1957 ◽  
Vol 12 ◽  
pp. 145-146 ◽  
Author(s):  
Albert Nijenhuis
Keyword(s):  
Vector Space ◽  
Lie Group ◽  
Holonomy Group ◽  
Direct Proof ◽  
Structure Group ◽  
Fiber Bundles ◽  
General Fiber ◽  
Holonomy Groups

In a recent paper [1], H. Ozeki has extended the author’s results [2] on local and infinitesimal holonomy groups for connections in linear fiber bundles (whose fiber is a vector space) to general fiber bundles whose structure group is a Lie group. Ozeki’s Lemma 7 (corresponding to the author’s Lemma 5.2) appears to be rather crucial in the development, while the proof is somewhat involved.—This note intends to present a more direct proof of the lemma, stating a case in which the local holonomy group H*(x)and the infinitesimal holonomy group H’(x)coincide:

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.

scholarly journals Most finitely generated subgroups of infinite unitriangular matrices are free

2002 ◽  
Vol 66 (3) ◽  
pp. 419-423 ◽  
Author(s):  
Waldemar Hołubowski
Keyword(s):  
Finite Field ◽  
Free Groups ◽  
Finitely Generated ◽  
Infinite Dimensional ◽  
Suitable Sense ◽  
Countable Subgroup ◽  
Almost All ◽  
Free Subgroups

In this note we prove that the group G of infinite dimensional upper unitriangular matrices over a finite field contains an explicit countable subgroup ‘full’ of free subgroups. We deduce from this fact that, in a suitable sense, almost all k–generator subgroups of G are free groups of rank k.

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.

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