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Harmonicity and minimality of complex and quaternionic radial foliations
Abstract
We construct special classes of totally geodesic almost regular foliations, namely, complex radial foliations in Hermitian manifolds and quaternionic radial foliations in quaternionic Kähler manifolds, and we give criteria for their harmonicity and minimality. Then examples of these foliations on complex and quaternionic space forms, which are harmonic and minimal, are presented.
Let Gk,n be the Grassmannian of oriented subspaces of dimension k of Rn with its canonical Riemannian metric. We study the energy of maps assigning to each P∈Gk,n a unit vector normal to P. They are sections of a sphere bundle Ek,n1 over Gk,n. The octonionic double and triple cross products induce in a natural way such sections for k=2, n=7 and k=3, n=8, respectively. We prove that they are harmonic maps into Ek,n1 endowed with the Sasaki metric. This, together with the well-known result that Hopf vector fields on odd dimensional spheres are harmonic maps into their unit tangent bundles, allows us to conclude that all unit normal sections of the Grassmannians associated with cross products are harmonic. In a second instance we analyze the energy of maps assigning an orthogonal complex structure JP on P⊥ to each P∈G2,8. We prove that the one induced by the octonionic triple product is a harmonic map into a suitable sphere bundle over G2,8. This generalizes the harmonicity of the canonical almost complex structure of S6.
Let G(k,n) be the Grassmannian of oriented subspaces of dimension k of R^n with its canonical Riemannian metric. We study the energy of maps assigning to each P \in G(k,n) a unit vector normal to P. They are sections of a sphere bundle E_{k,n}^1 over G(k,n). The octonionic double and triple cross products induce in a natural way such sections for k=2, n=7 and k=3, n=8, respectively. We prove that they are harmonic maps into E_{k,n}^1 endowed with the Sasaki metric. This, together with the well-known result that Hopf vector fields on odd dimensional spheres are harmonic maps into their unit tangent bundles, allows us to conclude that all unit normal sections of the Grassmannians associated with cross products are harmonic. In a second instance we analyze the energy of maps assigning an orthogonal complex structure J(P) on P^{\bot} to each P\in G(2,8). We prove that the one induced by the octonionic triple product is a harmonic map into a suitable sphere bundle over G(2,8). This generalizes the harmonicity of the canonical almost complex structure of S^6.
For closed and connected subgroups G of SO(n), we study the energy functional on the space of G-structures of a (compact) Riemannian manifold M, where G-structures are considered as sections of the quotient bundle O(M)/G. Then, we deduce the corresponding first and second variation formulae and the characterising conditions for critical points by means of tools closely related with the study of G-structures. In this direction, we show the role in the energy functional played by the intrinsic torsion of the G-structure. Moreover, we analyse the particular case G=U(n) for even-dimensional manifolds. This leads to the study of harmonic almost Hermitian manifolds and harmonic maps from M into O(M)/U(n). Comment: 27 pages, minor corrections
We consider the energy of smooth generalized distributions and also of
singular foliations on compact Riemannian manifolds for which the set of their
singularities consists of a finite number of isolated points and of pairwise
disjoint closed submanifolds. We derive a lower bound for the energy of all
$q$-dimensional almost regular distributions, for each $q< \dim M,$ and find
several examples of foliations which minimize the energy functional over
certain sets of smooth generalized distributions.
We study the geometry of a $G$--structure $P$ inside the oriented orthonornal
frame bundle $SO(M)$ over oriented Riemannian manifold $M$. We assume the
quotient $SO(n)/G$, where $n=\dim M$, is a natural homogeneous space and we
equip $SO(M)$ with the natural Riemannian structure induced from the structure
on $M$ and the Killing form of $SO(n)$. We show, in particular, that minimality
of $P$ is equivalent to harmonicity of a induced section of the homogeneous
bundle $SO(M)\times_{SO(n)}SO(n)/G$ with the modified Riemannian metric on $M$
and the minimality of the image of this section. We apply obtained results to
the case of almost product structures, i.e. structures induced by plane fields,
and to almost hermitian structures.
We consider a q-dimensional distribution as a section of the Grassmannian bundle G(q)(M-n) of q-planes and we derive, in terms of the intrinsic torsion of the corresponding S(O(q) x O(n-q))-structure, the conditions that this map must satisfy in order to be critical for the functionals energy and volume. Using this it is shown that invariant Riemannian foliations of homogeneous Riemannian manifolds which are transversally symmetric determine harmonic maps and minimal immersions. In particular, canonical homogeneous fibrations on rank one normal homogeneous spaces or on compact irreducible 3-symmetric spaces provide many examples of harmonic maps and minimal immersions of compact Riemannian manifolds.
We extend theorems of É. Cartan, Nomizu, Münzner, Q.M. Wang, and Ge–Tang on isoparametric functions to transnormal functions on a general Riemannian manifold. We show that if a complete Riemannian manifold M admits a transnormal function, then M is diffeomorphic to either a vector bundle over a submanifold, or a union of two disk bundles over two submanifolds. Moreover, a singular level set Q is austere and minimal, if exists, and generic level sets are tubes over Q. We give a criterion for a transnormal function to be an isoparametric function.
We present new examples of harmonic and minimal unit vector fields. These are radial vector fields on tubular neighbourhoods
about points and submanifolds in two-point homogeneous spaces and harmonic manifolds, and about characteristic curves in Sasakian
space forms.
Let D D be an arbitrary set of C ∞ {C^\infty } vector fields on the C ∞ {C^\infty } manifold M M . It is shown that the orbits of D D are C ∞ {C^\infty } submanifolds of M M , and that, moreover, they are the maximal integral submanifolds of a certain C ∞ {C^\infty } distribution P D {P_D} . (In general, the dimension of P D ( m ) {P_D}(m) will not be the same for all m ∈ M m \in M .) The second main result gives necessary and sufficient conditions for a distribution to be integrable. These two results imply as easy corollaries the theorem of Chow about the points attainable by broken integral curves of a family of vector fields, and all the known results about integrability of distributions (i.e. the classical theorem of Frobenius for the case of constant dimension and the more recent work of Hermann, Nagano, Lobry and Matsuda). Hermann and Lobry studied orbits in connection with their work on the accessibility problem in control theory. Their method was to apply Chow’s theorem to the maximal integral submanifolds of the smallest distribution Δ \Delta such that every vector field X X in the Lie algebra generated by D D belongs to Δ \Delta (i.e. X ( m ) ∈ Δ ( m ) X(m) \in \Delta (m) for every m ∈ M m \in M ). Their work therefore requires the additional assumption that Δ \Delta be integrable. Here the opposite approach is taken. The orbits are studied directly, and the integrability of Δ \Delta is not assumed in proving the first main result. It turns out that Δ \Delta is integrable if and only if Δ = P D \Delta = {P_D} , and this fact makes it possible to derive a characterization of integrability and Chow’s theorem. Therefore, the approach presented here generalizes and unifies the work of the authors quoted above.
We consider the energy (or the total bending) of unit vector fields oncompact Riemannian manifolds for which the set of its singularitiesconsists of a finite number of isolated points and a finite number ofpairwise disjoint closed submanifolds. We determine lower bounds for theenergy of such vector fields on general compact Riemannian manifolds andin particular on compact rank one symmetric spaces. For this last classof spaces, we compute explicit expressions for the total bending whenthe unit vector field is the gradient field of the distance function toa point or to special totally geodesic submanifolds (i.e., for radialunit vector fields around this point or these submanifolds).
We consider an oriented distribution as a section of the corresponding Grassmann bundle and, by computing the tension of this
map for conveniently chosen metrics, we obtain the conditions which the distribution must satisfy in order to be critical
for the functionals related to the volume or the energy of the map. We show that the three-dimensional distribution ofS
4m+3 tangent to the quaternionic Hopf fibration defines a harmonic map and a minimal immersion and we extend these results to
more general situations coming from 3-Sasakian and quaternionic geometry.
On etablit les proprietes de base des fonctions transnormales et on demontre le lemme du tube. On demontre que: soit M une variete de Riemann lisse, complete, connexe et f une fonction transnormale sur M. Alors: a) les varietes focales de f sont des sous-varietes lisses de M; b) chaque ensemble de niveau regulier de f est un tube sur d'autres des varietes focales
We show how the equations for harmonic maps into homogeneous spaces generalize to harmonic sections of homogeneous fibre bundles. Surprisingly, the generalization does not explicitly involve the curvature of the bundle. However, a number of special cases of the harmonic section equations (including the new condition of super-flatness) are studied in which the bundle curvature does appear. Some examples are given to illustrate these special cases in the non-flat environment. The bundle in question is the twistor bundle of an even-dimensional Riemannian manifold M whose sections are the almost-Hermitian structures of M.
The energy of Riemannian almost-product structure P is measured by forming the Dirichlet integral of the associated Gauss section γ, and P is decreed harmonic if γ criticalizes the energy functional when restricted to the submanifold of sections of the Grassman bundle. Euler-Lagrange equations are obtained, and geometrically transformed in the special case when P is totally geodesic. These are seen to generalize the Yang-Mills equations, and generalizations of the self-duality and anti-self-duality conditions are suggested. Several applications are then described. In particular, it is considered whether integrability of P is a necessary condition for γ to be harmonic.
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.
In this paper we give characterizations of quaternionic space forms in the class of quaternionic Kähler manifolds in terms of the curvature tensor and the extrinsic shape of geodesics on geodesic spheres.
Energy of generalized distributions, Differential Geom CrossrefGoogle Scholar
- J C Gonzálezdávila