scholarly journals Nonnegatively curved 5–manifolds with almost maximal symmetry rank

2014 ◽  
Vol 18 (3) ◽  
pp. 1397-1435 ◽  
Author(s):  
Fernando Galaz-Garcia ◽  
Catherine Searle
Keyword(s):  
Maximal Symmetry ◽  
Symmetry Rank

Torus actions, maximality, and non-negative curvature

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Christine Escher ◽  
Catherine Searle
Keyword(s):  
Negative Curvature ◽  
Torus Action ◽  
Maximal Symmetry ◽  
Curved Manifolds ◽  
Negatively Curved ◽  
Product Of Spheres ◽  
Rank Conjecture ◽  
Simply Connected ◽  
Symmetry Rank ◽  
Special Case

Abstract Let ℳ 0 n {\mathcal{M}_{0}^{n}} be the class of closed, simply connected, non-negatively curved Riemannian n-manifolds admitting an isometric, effective, isotropy-maximal torus action. We prove that if M ∈ ℳ 0 n {M\in\mathcal{M}_{0}^{n}} , then M is equivariantly diffeomorphic to the free, linear quotient by a torus of a product of spheres of dimensions greater than or equal to 3. As a special case, we then prove the Maximal Symmetry Rank Conjecture for all M ∈ ℳ 0 n {M\in\mathcal{M}_{0}^{n}} . Finally, we show the Maximal Symmetry Rank Conjecture for simply connected, non-negatively curved manifolds holds for dimensions less than or equal to 9 without additional assumptions on the torus action.

scholarly journals Non-negatively Curved 6-Manifolds with Almost Maximal Symmetry Rank

2018 ◽  
Vol 29 (1) ◽  
pp. 1002-1017
Author(s):  
Christine Escher ◽  
Catherine Searle
Keyword(s):  
Maximal Symmetry ◽  
Negatively Curved ◽  
Symmetry Rank

scholarly journals Positively curved manifolds with maximal symmetry-rank

1994 ◽  
Vol 91 (1-3) ◽  
pp. 137-142 ◽  
Author(s):  
Karsten Grove ◽  
Catherine Searle
Keyword(s):  
Maximal Symmetry ◽  
Curved Manifolds ◽  
Positively Curved ◽  
Symmetry Rank

Some variational principles associated with ODEs of maximal symmetry. Part 2: The general case

2018 ◽  
Vol 24 (2) ◽  
pp. 175-183
Author(s):  
Jean-Claude Ndogmo
Keyword(s):  
Explicit Form ◽  
Nonlinear Equations ◽  
Variational Principles ◽  
Symmetry Algebra ◽  
First Integrals ◽  
Maximal Symmetry ◽  
General Order ◽  
Linear And Nonlinear ◽  
Variational Symmetries ◽  
Variational Symmetry

Abstract Variational and divergence symmetries are studied in this paper for the whole class of linear and nonlinear equations of maximal symmetry, and the associated first integrals are given in explicit form. All the main results obtained are formulated as theorems or conjectures for equations of a general order. A discussion of the existence of variational symmetries with respect to a different Lagrangian, which turns out to be the most common and most readily available one, is also carried out. This leads to significantly different results when compared with the former case of the transformed Lagrangian. The latter analysis also gives rise to more general results concerning the variational symmetry algebra of any linear or nonlinear equations.

scholarly journals DUALITY INVARIANCE OF COSMOLOGICAL SOLUTIONS WITH TORSION

1999 ◽  
Vol 14 (31) ◽  
pp. 4953-4966 ◽  
Author(s):  
DEBASHIS GANGOPADHYAY ◽  
SOUMITRA SENGUPTA
Keyword(s):  
Physical Meaning ◽  
Tensor Field ◽  
Antisymmetric Tensor ◽  
Dilaton Field ◽  
Maximal Symmetry ◽  
Killing Vectors ◽  
Cosmological Solutions ◽  
Antisymmetric Tensor Field ◽  
Vacuum Solutions

We show that for a string moving in a background consisting of maximally symmetric gravity, dilaton field and second rank antisymmetric tensor field, the O(d) ⊗ O(d) transformation on the vacuum solutions gives inequivalent solutions that are not maximally symmetric. We then show that the usual physical meaning of maximal symmetry can be made to remain unaltered even if torsion is present and illustrate this through two toy models by determining the torsion fields, the metric and Killing vectors. Finally we show that under the O(d) ⊗ O(d) transformation this generalized maximal symmetry can be preserved under certain conditions. This is interesting in the context of string related cosmological backgrounds.

Free Extensions of Chiral Polytopes

1995 ◽  
Vol 47 (3) ◽  
pp. 641-654 ◽  
Author(s):  
Egon Schulte ◽  
Asia Ivić Weiss
Keyword(s):  
Classical Theory ◽  
Convex Polytope ◽  
Regular Polytopes ◽  
Geometric Structures ◽  
Maximal Symmetry ◽  
Abstract Polytopes ◽  
Chiral Polytopes ◽  
Classical Notion ◽  
Extension Result ◽  
Amalgamated Product

AbstractAbstract polytopes are discrete geometric structures which generalize the classical notion of a convex polytope. Chiral polytopes are those abstract polytopes which have maximal symmetry by rotation, in contrast to the abstract regular polytopes which have maximal symmetry by reflection. Chirality is a fascinating phenomenon which does not occur in the classical theory. The paper proves the following general extension result for chiral polytopes. If 𝒦 is a chiral polytope with regular facets 𝓕 then among all chiral polytopes with facets 𝒦 there is a universal such polytope 𝓟, whose group is a certain amalgamated product of the groups of 𝒦 and 𝓕. Finite extensions are also discussed.

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