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 r-Harmonic and Complex Isoparametric Functions on the Lie Groups $${{\mathbb {R}}}^m \ltimes {{\mathbb {R}}}^n$$ and $${{\mathbb {R}}}^m \ltimes \mathrm {H}^{2n+1}$$

2020 ◽  
Vol 58 (4) ◽  
pp. 477-496
Author(s):  
Sigmundur Gudmundsson ◽  
Marko Sobak
Keyword(s):  
Heisenberg Group ◽  
Lie Groups ◽  
Lie Group ◽  
Riemannian Manifolds ◽  
Harmonic Functions ◽  
Semidirect Products ◽  
Simply Connected ◽  
Isoparametric Functions ◽  
General Method

Abstract In this paper we introduce the notion of complex isoparametric functions on Riemannian manifolds. These are then employed to devise a general method for constructing proper r-harmonic functions. We then apply this to construct the first known explicit proper r-harmonic functions on the Lie group semidirect products $${{\mathbb {R}}}^m \ltimes {{\mathbb {R}}}^n$$ R m ⋉ R n and $${{\mathbb {R}}}^m \ltimes \mathrm {H}^{2n+1}$$ R m ⋉ H 2 n + 1 , where $$\mathrm {H}^{2n+1}$$ H 2 n + 1 denotes the classical $$(2n+1)$$ ( 2 n + 1 ) -dimensional Heisenberg group. In particular, we construct such examples on all the simply connected irreducible four-dimensional Lie groups.

scholarly journals Representations of complex semisimple Lie groups and Lie algebras

1966 ◽  
Vol 72 (3) ◽  
pp. 522-526 ◽  
Author(s):  
K. R. Parthasarathy ◽  
R. Ranga Rao ◽  
V. S. Varadarajan
Keyword(s):  
Lie Groups ◽  
Lie Algebras ◽  
Semisimple Lie Groups ◽  
Complex Semisimple

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.

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