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 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 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.

Note on Best Approximation of |x|

1979 ◽  
Vol 22 (3) ◽  
pp. 363-366
Author(s):  
Colin Bennett ◽  
Karl Rudnick ◽  
Jeffrey D. Vaaler
Keyword(s):  
General Problem ◽  
Uniform Approximation ◽  
Best Approximation ◽  
Best Uniform Approximation ◽  
Linear Fractional Transformations ◽  
Symmetric Complex ◽  
Image Position ◽  
Complex Valued ◽  
Special Case

In this note the best uniform approximation on [—1,1] to the function |x| by symmetric complex valued linear fractional transformations is determined. This is a special case of the more general problem studied in [1]. Namely, for any even, real valued function f(x) on [-1,1] satsifying 0 = f ( 0 ) ≤ f (x) ≤ f (1) = 1, determine the degree of symmetric approximationand the extremal transformations U whenever they exist.

scholarly journals On the Non-Existence of Injective Near-Ring Modules

1977 ◽  
Vol 20 (1) ◽  
pp. 17-23 ◽  
Author(s):  
Bernhard Banaschewski ◽  
Evelyn Nelson
Keyword(s):  
Natural Notion ◽  
Injective Hulls ◽  
Special Case

Associated with any near-ring R, and any set S of distributive elements of R, one has a natural notion of S-distributive R-modules, analogous to that of modules over rings and including the latter as special case (Frohlich [6]; we recall the details in Section 1). Since near-rings can be viewed as slightly deficient rings, it makes sense to enquire whether such near-ring modules share with modules over rings the familiar and important property of having injective hulls.

scholarly journals On Fε2-planar mappings with function ε of (pseudo-)Riemannian manifolds

Filomat ◽  
2017 ◽  
Vol 31 (9) ◽  
pp. 2683-2689 ◽  
Author(s):  
Hana Chudá ◽  
Nadezda Guseva ◽  
Patrik Peska
Keyword(s):  
Partial Differential Equations ◽  
Riemannian Manifolds ◽  
Initial Conditions ◽  
Riemannian Metrics ◽  
Cauchy Type ◽  
Zero Function ◽  
Projective Equivalence ◽  
Type Form ◽  
Planar Mapping ◽  
Special Case

In this paper we study special mappings between n-dimensional (pseudo-) Riemannian manifolds. In 2003 Topalov introduced PQ?-projectivity of Riemannian metrics, with constant ? ? 0,1 + n. These mappings were studied later by Matveev and Rosemann and they found that for ? = 0 they are projective. These mappings could be generalized for case, when ? will be a function on manifold. We show that PQ?- projective equivalence with ? is a function corresponds to a special case of F-planar mapping, studied by Mikes and Sinyukov (1983) with F = Q. Moreover, the tensor P is derived from the tensor Q and non-zero function ?. We assume that studied mappings will be also F2-planar (Mikes 1994). This is the reason, why we suggest to rename PQ? mapping as F?2. For these mappings we find the fundamental partial differential equations in closed linear Cauchy type form and we obtain new results for initial conditions.

scholarly journals Proper r-harmonic functions from Riemannian manifolds

2019 ◽  
Vol 57 (1) ◽  
pp. 217-223 ◽  
Author(s):  
Sigmundur Gudmundsson ◽  
Marko Sobak
Keyword(s):  
Lie Groups ◽  
Riemannian Manifolds ◽  
Harmonic Functions ◽  
New Method ◽  
Semisimple Lie Groups ◽  
Complex Valued

AbstractWe introduce a new method for constructing complex-valued r-harmonic functions on Riemannian manifolds. We then apply this for the important semisimple Lie groups $$\mathbf{SO }(n)$$SO(n), $$\mathbf{SU }(n)$$SU(n), $$\mathbf{Sp }(n)$$Sp(n), $$\mathbf{SL }_{n}({\mathbb {R}})$$SLn(R), $$\mathbf{Sp }(n,{\mathbb {R}})$$Sp(n,R), $$\mathbf{SU }(p,q)$$SU(p,q), $$\mathbf{SO }(p,q)$$SO(p,q), $$\mathbf{Sp }(p,q)$$Sp(p,q), $$\mathbf{SO }^*(2n)$$SO∗(2n) and $$\mathbf{SU }^*(2n)$$SU∗(2n).

scholarly journals Harmonic morphisms with one-dimensional fibres on conformally-flat Riemannian manifolds

2008 ◽  
Vol 145 (1) ◽  
pp. 141-151 ◽  
Author(s):  
RADU PANTILIE
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Harmonic Morphisms ◽  
Two Dimensional ◽  
Conformally Flat ◽  
Harmonic Morphism ◽  
One Dimensional ◽  
Dimension 2 ◽  
Real Analytic ◽  
Flat Riemannian Manifold

AbstractWe classify the harmonic morphisms with one-dimensional fibres (1) from real-analytic conformally-flat Riemannian manifolds of dimension at least four (Theorem 3.1), and (2) between conformally-flat Riemannian manifolds of dimensions at least three (Corollaries 3.4 and 3.6).Also, we prove (Proposition 2.5) an integrability result for any real-analytic submersion, from a constant curvature Riemannian manifold of dimensionn+2 to a Riemannian manifold of dimension 2, which can be factorised as ann-harmonic morphism with two-dimensional fibres, to a conformally-flat Riemannian manifold, followed by a horizontally conformal submersion, (n≥4).

Sign in / Sign up

Export Citation Format

Share Document