Abstract

The main object in this monograph are harmonic vector fields on Riemannian manifolds. Let (M,g) be an n-dimensional Riemannian manifold, and S(M) be its tangent sphere bundle, where S(M) is endowed with a Riemannian metric G_s (the Sasaki metric) associated to the Riemannian metric g on M. Thus for a smooth unit tangent field X:M→S(M) from (M,g) into (S(M),G_s), we can consider the Dirichlet energy E(X)=1/2∫_M ∥dX∥2 dv_g. A harmonic vector field is a critical point of E(X) for any smooth 1-parameter variation of X through unit tangent vector fields. In terms of the Euler-Lagrange equation, a harmonic vector field is a smooth solution of the nonlinear elliptic PDE system ΔgX=∥∇X∥^2 X, where Δ_g is the rough Laplacian. Harmonic vector fields are not necessarily harmonic maps except when the curvature condition trace_g {R(∇⋅X,X)⋅}=0 is satisfied. The resulting theory of harmonic vector fields is similar in many respects to the more consolidated theory of harmonic maps yet presents new and intriguing aspects captured in a rapidly growing specific literature. The monograph (over 500 pages) is self-contained in the field of harmonic vector fields, and consists of eight chapters, as follows: (1) Geometry of the tangent bundle, (2) Harmonic vector fields, (3) Harmonicity and stability, (4) Harmonicity and contact metric structures, (5) Harmonicity with respect to g-natural metrics, (6) The energy of sections, (7) Harmonic vector fields in CR geometry, and (8) Lorentz geometry and harmonic vector fields.

Autore Pugliese

• Perrone Domenico

Tutti gli autori

• S. Dragomir , D. Perrone

Titolo volume/Rivista

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Anno di pubblicazione

2011

ISSN

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ISBN

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