Distributions on Riemannian manifolds, which are harmonic maps

Abstract

We find new examples of harmaonic maps between compact Riemannian manifolds. A section of a Riemannian fibration is called harmonic if it is harmonic as a map from the base manifold into the total space. When the fibres are totally geodesic, the Euler-Lagrange equation for such sections is formulated. In the case of distributions, which are sections of a Grassmannian bundle, this formula is described in terms of the geometry of base manifolds. Examples of harmonic distributions are constructed when the base manifolds are homogeneous spaces and the integral submanifolds are totally geodesic. In particular, we show all the generalized Hopf-fibrations define harmonic maps into the Grassmannian bundles with the standard metric. Introduction.Let M and N be complete Riemannian manifolds. Assume M is compact. A smooth map f : M → N is called harmonic if it is a critical point of the energy functional. If N has non-positive curvature, then there exists a harmonic map in each homotopy class. However, if N has positive curvature, this is no longer true and it always has been meaningful to find examples of harmonic map between compact manifolds with nonnegative curvature. In this paper, we produce some new type of harmonic maps between compact Riemannian manifolds.We consider the case when N is a fibre bundle over M and f : M → N is a smooth map which happens to be a section of this fibration. If N is a vector bundle over M with a connection type metric, then f is a harmonic map if and only if it is a parallel section. We will consider the case when the fibres are totally geodesic compact submanifolds, and hence N is also a compact Riemannian manifold. In particular, if N is a Grassmannian bundle associated to the orthogonal bundle O(M), then the map f defines a smooth distribution on M. A distribution on a Riemannian manifold M will be called harmonic if it is harmonic as a map from M into the total space of the Grassmannian bundle over M. We will find the Euler-Lagrange equation of a harmonic map f in a general setting and find a solution to this equation to construct examples of harmonic distributions.The definition of harmonic distributions, or harmonic sections in general, is used by other authors [7], [8], [9], [13] in a similar context but for a different concept. They call a section harmonic if its vertical energy is stationary with respect to vertical variations. If one is looking for a better section, this notion of harmonic section makes more sense. A variation

Harmonicity and minimality of distributions on Riemannian manifolds via the intrinsic torsion

“…We shall show in Theorem 3.2 that the Riemannian metrics g K and g S on G or q (M ) determine the same Riemannian structure. Hence, studies of the harmonicity for distributions made by viewing them as maps into (G q (M ), g K ), as in [5] and [28], among others; or for oriented distributions by viewing them as maps into (G or q (M ), i * g S ), as in [4], [8], [20] and [22], yield the same theory.…”

Energy of generalized distributions

“…It makes π : (G(M r ), g K ) → (M r , g Mr ) a Riemannian submersion with totally geodesic fibres, where g Mr denotes the induced metric by g on M r . The energy of a q-dimensional regular distribution σ is defined in [8] (see also [6] and [18]) as the energy of the map σ : (M, g) → (G q (M ), g K ). An equivalent definition for oriented regular distributions, considered as sections of the bundle of unit decomposable q-vectors equipped with the generalized Sasaki metric, is given in [4] and [7], among others.…”

Two notes on harmonic distributions

“…Then σ is spanned by vector X. One can show that σ is harmonic with respect to standard inner product g on S 3 [4,3]. Suppose there exists function µ such that σ is harmonic with respect to e 2µ g, where µ is not a constant.…”

“…(1) generalized Hopf fibrations [3,4], (2) characteristic distribution of a contact structure Notice that example (2) is a special case of example (4) since Reeb vector field of a contact structure is unit harmonic.…”

“…(1) generalized Hopf fibrations [3,4], (2) characteristic distribution of a contact structure [4], (3) invariant distributions on compact semisimple Lie groups [2]. (4) distributions orthogonal to harmonic unit vector fields [4].…”