Sorin Dragomir

Harmonic Vector Fields

Publisher Summary This chapter discusses fundamental matters such as the Dirichlet energy and tension tensor of a unit tangent vector field on a Riemannian manifold, first and second variation formulae, and the harmonic vector fields system. The study of the weak solutions to this system (existence and local properties) is missing from the present day mathematical literature. Various instances are investigated where harmonic vector fields occur and to generalizations. Any unit vector field that is a harmonic map is also a harmonic vector field. The study of harmonic map system is more appropriate on a Hermitian manifold and that results in Hermitian harmonic maps to be useful in studying rigidity of complete Hermitian manifolds.

Yang–Mills fields on CR manifolds

We study pseudo Yang–Mills fields on a compact strictly pseudoconvex CR manifold M, i.e., the critical points of the functional PYM(D)=12∫M∥πHRD∥2θ∧(dθ)n, where D is a connection in a Hermitian CR holomorphic vector bundle (E,h)→M. Let Ω={φ<0}⊂Cn be a smoothly bounded strictly pseuodoconvex domain and g the Bergman metric on Ω. We show that boundary values Db of Yang–Mills fields D on (Ω,g) are pseudo Yang–Mills fields on ∂Ω, provided that iTRDb=0 and iNRD=0 on H(∂Ω). If S1→C(M)→πM is the canonical circle bundle and π*D is a Yang–Mills field with respect to the Fefferman metric Fθ of (M,θ) then D is a pseudo Yang–Mills field on M. The Yang–Mills equations δπ*DRπ*D=0 project on the Euler–Lagrange equations δbDRD=0 of the variational principle δPYM(D)=0, provided that iTRD=0. When M has vanishing pseudohermitian Ricci curvature the pullback π*D of the (CR invariant) Tanaka connection D of (E,h) is a Yang–Mills field on C(M). We derive the second variation formula {d2PYM(Dt)∕dt2}t=0=∫M⟨SbD(φ),φ⟩θ∧(dθ)n, Dt=D...

On Transversally Holomorphic Maps of Kählerian Foliations

AbstractAny transversally holomorphic foliated map

Mixed gravitational field equations on globally hyperbolic spacetimes

\varphi :(M,\mathcal{F}) \to (M\prime ,\mathcal{F}\prime )

Propagation of singularities along characteristics of Maxwell's equations

of Kählerianfoliations with

On Lewy's unsolvability phenomenon

\mathcal{F}