Applications of harmonic morphisms to gravity

Abstract

We i n troduce the notion of gravity coupled to a horizontally conformal submersion as a modication of the well-known concept of gravity coupled to a harmonic map, t h us obtaining a coupled gravity system with more geometric avour. By using integral techniques we determine the necessary conditions for coupling and cosmological constants. Finally, in the context of higher dimensional gravitation theory, we show that harmonic morphisms provide a natural ansatz to trigger spontaneous splitting and reduction of the gravity system coupled to a harmonic map on

Semi-Slant Submersions from Almost Product Riemannian Manifolds

“…Then the second fundamental form of π is given by (10) p∇π˚qpX, Y q " ∇ π˚X π˚Y´π˚p∇ X Y q for X, Y P ΓpT M q, where we denote conveniently by ∇ the Levi-Civita connections of the metrics g M and g N . Recall that π is said to be harmonic if tracep∇π˚q " 0 and π is called a totally geodesic…”

“…Riemannian submersion ( [5], [11]), slant submersion ( [6], [12], [13]), almost Hermitian submersion [16], quaternionic submersion [7], etc. As we know, Riemannian submersions are related with physics and have their applications in the Yang-Mills theory ( [3], [17]), Kaluza-Klein theory ( [2], [8]), semi-invariant submersion( [14]), supergravity and superstring theories ( [9], [10]), etc. In [15], the author studied the slant and semi-slant submanifolds of an almost product Riemannian manifold.…”

“…For X, Y P Γpkerπ˚q, using (2), (10) and (12) we have p∇π˚qpX, Y q "´π˚p∇ X Y q "´π˚pF ∇ X pφY`ωY q.…”

Pointwise Slant Submersions

“…Riemannian submersions have several applications in mathematical physics. Indeed, Riemannian submersions have their applications in the Yang-Mills theory ( [2,24]), Kaluza-Klein theory ( [3,15]), supergravity and superstring theories ( [16,18]), etc. Later such submersions were considered between manifolds with differentiable structures, see: [11].…”

H-v-Semi-Slant Submersions From Almost Quaternionic Hermitian Manifolds

“…As we know, Riemannian submersions are related with physics and have their applications in the Yang-Mills theory ( [4], [25]), Kaluza-Klein theory ( [5], [12]), Supergravity and superstring theories ( [13], [15]), etc. And the quaternionic Kähler manifolds have applications in physics as the target spaces for nonlinear σ-models with supersymmetry [7].…”