Infinitesimal transformations of a symmetric Riemannian space of the first class

Infinitesimal rotary transformation

Filomat ◽

10.2298/fil1904153r ◽

2019 ◽

Vol 33 (4) ◽

pp. 1153-1157

Author(s):

Lenka Rýparová ◽

Josef Mikes

Keyword(s):

Riemannian Space ◽

Basic Equations ◽

Fundamental Equations ◽

Principal Parts ◽

Infinitesimal Transformations ◽

Riemannian Spaces

The paper is devoted to further study of a certain type of infinitesimal transformations of twodimensional (pseudo-) Riemannian spaces, which are called rotary. Aninfinitesimal transformation is called rotary if it maps any geodesic on (pseudo-) Riemannian space onto an isoperimetric extremal of rotation in their principal parts on (pseudo-) Riemannian space. We study basic equations of the infinitesimal rotary transformations in detail and obtain the simpler fundamental equations of these transformations.

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Projective Collineations of Decomposable Spacetimes Generated by the Lie Point Symmetries of Geodesic Equations

Symmetry ◽

10.3390/sym13061018 ◽

2021 ◽

Vol 13 (6) ◽

pp. 1018

Author(s):

Andronikos Paliathanasis

Keyword(s):

Differential Equations ◽

Riemannian Space ◽

Dimensional Subspace ◽

Geometric Construction ◽

Lie Point Symmetries ◽

Geodesic Equations ◽

Symmetries Of Differential Equations ◽

Projective Collineations

We investigate the relation of the Lie point symmetries for the geodesic equations with the collineations of decomposable spacetimes. We review previous results in the literature on the Lie point symmetries of the geodesic equations and we follow a previous proposed geometric construction approach for the symmetries of differential equations. In this study, we prove that the projective collineations of a n+1-dimensional decomposable Riemannian space are the Lie point symmetries for geodesic equations of the n-dimensional subspace. We demonstrate the application of our results with the presentation of applications.

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Uniqueness of Curvature Measures in Pseudo-Riemannian Geometry

Journal of Geometric Analysis ◽

10.1007/s12220-021-00702-4 ◽

2021 ◽

Author(s):

Andreas Bernig ◽

Dmitry Faifman ◽

Gil Solanes

Keyword(s):

Riemannian Manifolds ◽

Riemannian Geometry ◽

Riemannian Space ◽

Space Forms ◽

Type Formula ◽

Isometric Embeddings ◽

Curvature Measures ◽

Riemannian Space Forms

AbstractThe recently introduced Lipschitz–Killing curvature measures on pseudo-Riemannian manifolds satisfy a Weyl principle, i.e. are invariant under isometric embeddings. We show that they are uniquely characterized by this property. We apply this characterization to prove a Künneth-type formula for Lipschitz–Killing curvature measures, and to classify the invariant generalized valuations and curvature measures on all isotropic pseudo-Riemannian space forms.

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