Isometry groups of three‐dimensional Riemannian metrics

1992 ◽  
Vol 33 (1) ◽  
pp. 267-272 ◽  
Author(s):  
Carles Bona ◽  
Bartolomé Coll
Keyword(s):  
Three Dimensional ◽  
Riemannian Metrics ◽  
Isometry Groups

scholarly journals Dimension reduction for thin films prestrained by shallow curvature

2021 ◽  
Vol 477 (2247) ◽  
Author(s):  
Silvia Jiménez Bolaños ◽  
Marta Lewicka
Keyword(s):  
Dimension Reduction ◽  
Scaling Laws ◽  
Three Dimensional ◽  
Riemannian Metrics ◽  
Regular Solutions ◽  
Shell Models ◽  
Energy Scaling ◽  
New Energy ◽  
Monge Ampere ◽  
Equivalent Conditions

We are concerned with the dimension reduction analysis for thin three-dimensional elastic films, prestrained via Riemannian metrics with weak curvatures. For the prestrain inducing the incompatible version of the Föppl–von Kármán equations, we find the Γ -limits of the rescaled energies, identify the optimal energy scaling laws, and display the equivalent conditions for optimality in terms of both the prestrain components and the curvatures of the related Riemannian metrics. When the stretching-inducing prestrain carries no in-plane modes, we discover similarities with the previously described shallow shell models. In higher prestrain regimes, we prove new energy upper bounds by constructing deformations as the Kirchhoff–Love extensions of the highly perturbative, Hölder-regular solutions to the Monge–Ampere equation obtained by means of convex integration.

scholarly journals Diagonalization of three-dimensional pseudo-Riemannian metrics

2013 ◽  
Vol 74 ◽  
pp. 251-255 ◽  
Author(s):  
Oldřich Kowalski ◽  
Masami Sekizawa
Keyword(s):  
Three Dimensional ◽  
Riemannian Metrics

scholarly journals Left invariant Randers metrics of Berwald type on tangent Lie groups

2017 ◽  
Vol 15 (01) ◽  
pp. 1850015
Author(s):  
Farhad Asgari ◽  
Hamid Reza Salimi Moghaddam
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Three Dimensional ◽  
Riemannian Metrics ◽  
Randers Metric ◽  
Tangent Lie Group ◽  
Left Invariant Riemannian Metric ◽  
Simply Connected ◽  
Randers Metrics ◽  
Tangent Bundles

Let [Formula: see text] be a Lie group equipped with a left invariant Randers metric of Berward type [Formula: see text], with underlying left invariant Riemannian metric [Formula: see text]. Suppose that [Formula: see text] and [Formula: see text] are lifted Randers and Riemannian metrics arising from [Formula: see text] and [Formula: see text] on the tangent Lie group [Formula: see text] by vertical and complete lifts. In this paper, we study the relations between the flag curvature of the Randers manifold [Formula: see text] and the sectional curvature of the Riemannian manifold [Formula: see text] when [Formula: see text] is of Berwald type. Then we give all simply connected three-dimensional Lie groups such that their tangent bundles admit Randers metrics of Berwarld type and their geodesics vectors.

On the Prescribed Scalar and Zero Mean Curvature on 3-Dimensional Manifolds With Umbilic Boundary

2006 ◽  
Vol 6 (1) ◽  
Author(s):  
Mohameden Ould Ahmedou
Keyword(s):  
Mean Curvature ◽  
Three Dimensional ◽  
Riemannian Metrics ◽  
Dimensional Manifold ◽  
3 Dimensional ◽  
Prescribed Mean Curvature ◽  
Zero Mean Curvature ◽  
Boundary Mean Curvature

AbstractIn this paper we consider the existence and the compactness of Riemannian metrics of prescribed mean curvature and zero boundary mean curvature on a three dimensional manifold with umbilic boundary (M, g

scholarly journals Isometries of Riemannian and sub-Riemannian structures on three-dimensional Lie groups

2017 ◽  
Vol 25 (2) ◽  
pp. 99-135
Author(s):  
Rory Biggs
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Isometry Group ◽  
Three Dimensional ◽  
Isotropy Subgroup ◽  
Left Translation ◽  
Isometry Groups ◽  
Group Automorphism ◽  
Group Of Automorphisms ◽  
Simply Connected

Abstract We investigate the isometry groups of the left-invariant Riemannian and sub-Riemannian structures on simply connected three-dimensional Lie groups. More specifically, we determine the isometry group for each normalized structure and hence characterize for exactly which structures (and groups) the isotropy subgroup of the identity is contained in the group of automorphisms of the Lie group. It turns out (in both the Riemannian and sub-Riemannian cases) that for most structures any isometry is the composition of a left translation and a Lie group automorphism.

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