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A characterization of isometries between Riemannian manifolds by using development along geodesic triangles

Petri Kokkonen (2012)

Archivum Mathematicum

In this paper we characterize the existence of Riemannian covering maps from a complete simply connected Riemannian manifold ( M , g ) onto a complete Riemannian manifold ( M ^ , g ^ ) in terms of developing geodesic triangles of M onto M ^ . More precisely, we show that if A 0 : T | x 0 M T | x ^ 0 M ^ is some isometric map between the tangent spaces and if for any two geodesic triangles γ , ω of M based at x 0 the development through A 0 of the composite path γ · ω onto M ^ results in a closed path based at x ^ 0 , then there exists a Riemannian covering map...

A characterization of the Riemann extension in terms of harmonicity

Cornelia-Livia Bejan, Şemsi Eken (2017)

Czechoslovak Mathematical Journal

If ( M , ) is a manifold with a symmetric linear connection, then T * M can be endowed with the natural Riemann extension g ¯ (O. Kowalski and M. Sekizawa (2011), M. Sekizawa (1987)). Here we continue to study the harmonicity with respect to g ¯ initiated by C. L. Bejan and O. Kowalski (2015). More precisely, we first construct a canonical almost para-complex structure 𝒫 on ( T * M , g ¯ ) and prove that 𝒫 is harmonic (in the sense of E. García-Río, L. Vanhecke and M. E. Vázquez-Abal (1997)) if and only if g ¯ reduces to the...

A classification of locally homogeneous connections on 2-dimensional manifolds via group-theoretical approach

Oldřich Kowalski, Barbara Opozda, Zdeněk Vlášek (2004)

Open Mathematics

The aim of this paper is to classify (lócally) all torsion-less locally homogeneous affine connections on two-dimensional manifolds from a group-theoretical point of view. For this purpose, we are using the classification of all non-equivalent transitive Lie algebras of vector fields in ℝ2 according to P.J. Olver [7].

A local characterization of affine holomorphic immersions with an anti-complex and ∇-parallel shape operator

Maria Robaszewska (2002)

Annales Polonici Mathematici

We study the complex hypersurfaces f : M ( n ) n + 1 which together with their transversal bundles have the property that around any point of M there exists a local section of the transversal bundle inducing a ∇-parallel anti-complex shape operator S. We give a class of examples of such hypersurfaces with an arbitrary rank of S from 1 to [n/2] and show that every such hypersurface with positive type number and S ≠ 0 is locally of this kind, modulo an affine isomorphism of n + 1 .

Affine analogues of the Sasaki-Shchepetilov connection

Maria Robaszewska (2016)

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

For two-dimensional manifold M with locally symmetric connection ∇ and with ∇-parallel volume element vol one can construct a flat connection on the vector bundle TM ⊕ E, where E is a trivial bundle. The metrizable case, when M is a Riemannian manifold of constant curvature, together with its higher dimension generalizations, was studied by A.V. Shchepetilov [J. Phys. A: 36 (2003), 3893-3898]. This paper deals with the case of non-metrizable locally symmetric connection. Two flat connections on...

Affine reductive spaces

OLDRICH KOWALSKI (1979)

Beiträge zur Algebra und Geometrie = Contributions to algebra and geometry

Approximately Einstein ACH metrics, volume renormalization, and an invariant for contact manifolds

Neil Seshadri (2009)

Bulletin de la Société Mathématique de France

To any smooth compact manifold M endowed with a contact structure H and partially integrable almost CR structure J , we prove the existence and uniqueness, modulo high-order error terms and diffeomorphism action, of an approximately Einstein ACH (asymptotically complex hyperbolic) metric g on M × ( - 1 , 0 ) . We consider the asymptotic expansion, in powers of a special defining function, of the volume of M × ( - 1 , 0 ) with respect to g and prove that the log term coefficient is independent of J (and any choice of contact...

Basic equations of G -almost geodesic mappings of the second type, which have the property of reciprocity

Mića S. Stanković, Milan L. Zlatanović, Nenad O. Vesić (2015)

Czechoslovak Mathematical Journal

We study G -almost geodesic mappings of the second type θ π 2 ( e ) , θ = 1 , 2 between non-symmetric affine connection spaces. These mappings are a generalization of the second type almost geodesic mappings defined by N. S. Sinyukov (1979). We investigate a special type of these mappings in this paper. We also consider e -structures that generate mappings of type θ π 2 ( e ) , θ = 1 , 2 . For a mapping θ π 2 ( e , F ) , θ = 1 , 2 , we determine the basic equations which generate them.

Bi-Legendrian connections

Beniamino Cappelletti Montano (2005)

Annales Polonici Mathematici

We define the concept of a bi-Legendrian connection associated to a bi-Legendrian structure on an almost -manifold M 2 n + r . Among other things, we compute the torsion of this connection and prove that the curvature vanishes along the leaves of the bi-Legendrian structure. Moreover, we prove that if the bi-Legendrian connection is flat, then the bi-Legendrian structure is locally equivalent to the standard structure on 2 n + r .

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