scholarly journals Harmonic morphisms from even-dimensional hyperbolic spaces

2003 ◽  
Vol 92 (2) ◽  
pp. 246 ◽  
Author(s):  
Martin Svensson
Keyword(s):  
Riemannian Manifolds ◽  
Tangent Bundle ◽  
Hyperbolic Spaces ◽  
Harmonic Morphisms ◽  
Totally Geodesic ◽  
Complex Valued

In this paper we give a method for constructing complex valued harmonic morphisms in some pseudo-Riemannian manifolds using a parametrization of isotropic subbundles of the complexified tangent bundle. As a result we construct the first known examples of complex valued harmonic morphisms in real hyperbolic spaces of even dimension not equal to 4 which do not have totally geodesic fibres.

scholarly journals THE STRUCTURE OF HARMONIC MORPHISMS WITH TOTALLY GEODESIC FIBRES

2004 ◽  
Vol 06 (03) ◽  
pp. 419-430 ◽  
Author(s):  
M. T. MUSTAFA
Keyword(s):  
Riemannian Manifolds ◽  
Symmetric Spaces ◽  
Positive Curvature ◽  
Compact Manifolds ◽  
Harmonic Morphisms ◽  
Compact Type ◽  
Riemannian Submersions ◽  
Locally Symmetric Spaces ◽  
Totally Geodesic ◽  
Negatively Curved

The structure of local and global harmonic morphisms between Riemannian manifolds, with totally fibres, is investigated. It is shown that non-positive curvature of the domain obstructs the existence of global harmonic morphisms with totally geodesic fibres and the only such maps from compact Riemannian manifolds of non-positive curvature are, up to a homothety, totally geodesic Riemannian submersions. Similar results are obtained for local harmonic morphisms with totally geodesic fibres from open subsets of non-negatively curved compact and non-compact manifolds. During the course, we prove non-existence of submersive harmonic morphisms with totally geodesic fibres from some important domains, for instance from compact locally symmetric spaces of non-compact type and open subsets of symmetric spaces of compact type.

scholarly journals Complex-Valued (p, q)-Harmonic Morphisms from Riemannian Manifolds

2021 ◽  
Author(s):  
Elsa Ghandour ◽  
Sigmundur Gudmundsson
Keyword(s):  
Riemannian Manifolds ◽  
Harmonic Morphisms ◽  
Natural Notion ◽  
Complex Valued ◽  
Special Case

AbstractWe introduce the natural notion of (p, q)-harmonic morphisms between Riemannian manifolds. This unifies several theories that have been studied during the last decades. We then study the special case when the maps involved are complex-valued. For these we find a characterisation and provide new non-trivial examples in important cases.

scholarly journals Harmonic Morphisms Projecting Harmonic Functions to Harmonic Functions

2012 ◽  
Vol 2012 ◽  
pp. 1-8
Author(s):  
M. T. Mustafa
Keyword(s):  
Harmonic Function ◽  
Riemannian Manifolds ◽  
Harmonic Functions ◽  
Horizontal Component ◽  
Harmonic Morphisms

For Riemannian manifoldsMandN, admitting a submersionϕwith compact fibres, we introduce the projection of a function via its decomposition into horizontal and vertical components. By comparing the Laplacians onMandN, we determine conditions under which a harmonic function onU=ϕ−1(V)⊂Mprojects down, via its horizontal component, to a harmonic function onV⊂N.

scholarly journals Anti-invariant Riemannian submersions from cosymplectic manifolds onto Riemannian manifolds

Filomat ◽  
2015 ◽  
Vol 29 (7) ◽  
pp. 1429-1444 ◽  
Author(s):  
Cengizhan Murathan ◽  
Erken Küpeli
Keyword(s):  
Riemannian Manifolds ◽  
Riemannian Submersion ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Riemannian Submersions ◽  
Totally Geodesic ◽  
Cosymplectic Manifolds ◽  
Necessary And Sufficient ◽  
Decomposition Theorems ◽  
Characteristic Vector Field

We introduce anti-invariant Riemannian submersions from cosymplectic manifolds onto Riemannian manifolds. We survey main results of anti-invariant Riemannian submersions defined on cosymplectic manifolds. We investigate necessary and sufficient condition for an anti-invariant Riemannian submersion to be totally geodesic and harmonic. We give examples of anti-invariant submersions such that characteristic vector field ? is vertical or horizontal. Moreover we give decomposition theorems by using the existence of anti-invariant Riemannian submersions.

SEMI-INVARIANT RIEMANNIAN MAPS TO KÄHLER MANIFOLDS

2011 ◽  
Vol 08 (07) ◽  
pp. 1439-1454 ◽  
Author(s):  
BAYRAM ṢAHIN
Keyword(s):  
Riemannian Manifolds ◽  
Sufficient Conditions ◽  
Decomposition Theorem ◽  
Necessary And Sufficient Conditions ◽  
Classification Theorem ◽  
Totally Geodesic ◽  
Hermitian Manifolds ◽  
Almost Hermitian Manifolds ◽  
Riemannian Map ◽  
Necessary And Sufficient

This paper has two aims. First, we show that the usual notion of umbilical maps between Riemannian manifolds does not work for Riemannian maps. Then we introduce a new notion of umbilical Riemannian maps between Riemannian manifolds and give a method on how to construct examples of umbilical Riemannian maps. In the second part, as a generalization of CR-submanifolds, holomorphic submersions, anti-invariant submersions, invariant Riemannian maps and anti-invariant Riemannian maps, we introduce semi-invariant Riemannian maps from Riemannian manifolds to almost Hermitian manifolds, give examples and investigate the geometry of distributions which are arisen from definition. We also obtain a decomposition theorem and give necessary and sufficient conditions for a semi-invariant Riemannian map to be totally geodesic. Then we study the geometry of umbilical semi-invariant Riemannian maps and obtain a classification theorem for such Riemannian maps.

THE PRIME GEODESIC THEOREM AND QUANTUM MECHANICS ON FINITE VOLUME GRAPHS: A REVIEW

2001 ◽  
Vol 13 (12) ◽  
pp. 1459-1503 ◽  
Author(s):  
NORMAN E. HURT
Keyword(s):  
Finite Volume ◽  
Riemannian Manifolds ◽  
Riemann Surfaces ◽  
Three Dimensional ◽  
Hyperbolic Spaces ◽  
Quantum Graphs ◽  
Selberg Zeta Function ◽  
Ramanujan Graphs ◽  
Prime Geodesic Theorem ◽  
Ramanujan Conjecture

The prime geodesic theorem is reviewed for compact and finite volume Riemann surfaces and for finite and finite volume graphs. The methodology of how these results follow from the theory of the Selberg zeta function and the Selberg trace formula is outlined. Relationships to work on quantum graphs are surveyed. Extensions to compact Riemannian manifolds, in particular to three-dimensional hyperbolic spaces, are noted. Interconnections to the Selberg eigenvalue conjecture, the Ramanujan conjecture and Ramanujan graphs are developed.

On the geometry of harmonic morphisms

1990 ◽  
Vol 108 (3) ◽  
pp. 461-466 ◽  
Author(s):  
Sigmundur Gudmundsson
Keyword(s):  
Existence Results ◽  
Harmonic Morphisms ◽  
Connected Component ◽  
Euclidean Spaces ◽  
Conformal Maps ◽  
Orthogonal Projections ◽  
Totally Geodesic

AbstractLet π:M→B be a horizontally conformal submersion. We give necessary curvature conditions on the manifolds M and B, which lead to non-existence results for certain horizontally conformal maps, and harmonic morphisms. We then classify all such maps between open subsets of Euclidean spaces, which additionally have totally geodesic fibres and are horizontally homothetic. They are orthogonal projections on each connected component, followed by a homothety.

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