Totally geodesic maps into metric spaces

We prove that a totally geodesic map between a Riemannian manifold and a metric space can be represented as the composite of a totally geodesic map from a Riemannian manifold to a Finslerian manifold and a locally isometric embedding between metric spaces. As a corollary, we obtain the homotheticity of a totally geodesic map from an irreducible Riemannian manifold to an Alexandrov space of curvature bounded above. This is a generalization of the case between Riemannian manifolds. Mathematics Subject Classification (2000): 53C20, 53C22, 53C24 Received: 14 March 2002; in final form: 6 May 2002 / / Published online: 24 February 2003     

Homogeneous Spaces with Sections

We study homogeneous Riemannian manifolds all of whose geodesics can be mapped by some isometry into a fixed homogeneous, connected, totally geodesic submanifold, called section. We show that these spaces are locally symmetric if the section is two-dimensional and give non-symmetric counterexamples with higher-dimensional sections.Mathematics Subject Classification (2000): 53C35, 53C30, 53C22, 57S15     

Harmonic morphisms as unit normal bundles¶of minimal surfaces

Let be an isometric immersion between Riemannian manifolds and be the unit normal bundle of f. We discuss two natural Riemannian metrics on the total space and necessary and sufficient conditions on f for the projection map to be a harmonic morphism. We show that the projection map of the unit normal bundle of a minimal surface in a Riemannian manifold is a harmonic morphism with totally geodesic fibres. Received: 6 February 1999     

Fixed point and rigidity theorems for harmonic maps into NPC spaces

We show that harmonic maps from 2-dimensional Euclidean polyhedra to arbitrary NPC spaces are totally geodesic or constant depending on a geometric and combinatorial condition of the links of the 0-dimensional skeleton. Our method is based on a monotonicity formula rather than a codimension estimate of the singular set as developed by Gromov–Schoen or the mollification technique of Korevaar–Schoen.      

A Bochner Technique for Harmonic Morphisms

We establish a Weitzenböck formula for harmonic morphismsbetween Riemannian manifolds and show that under suitable curvatureconditions, such a map is totally geodesic. As an applicationof the Weitzenböck formula we obtain some non-existenceresults of a global nature for harmonic morphisms and totallygeodesic horizontally conformal maps between compact Riemannianmanifolds. In particular, it is shown that the only harmonicmorphisms from a Riemannian symmetric space of compact typeto a compact Riemann surface of genus at least 1 are the constantmaps.     

Finite energy and totally geodesic maps from locally symmetric spaces of finite volume

LetM=G/K be a locally symmetric space of finite volume and rank 2. We show that any map fromM of weighted finite energy in the sense of Saper can be deformed into a finite energy map. As a consequence such maps can be deformed into totally geodesic ones, and a geometric generalization of Margulis' superrigidity theorem is obtained.     

Minimal Homogeneous Submanifolds in Euclidean Spaces

We prove that minimal (extrinsically) homogeneous submanifolds of the Euclidean space are totally geodesic. As an application, we obtain that a complex homogeneous submanifold of C ^N must be totally geodesic.     

Nonlinear Markov operators associated with symmetric Markov kernels and energy minimizing maps between singular spaces

We present a theory of harmonic maps where the target is a complete geodesic space (N,d) of nonpositive curvature in the sense of A. D. Alexandrov and the domain is a measure space with a symmetric Markov kernel p on it. Our theory is a nonlinear generalization of the theory of symmetric Markov kernels and reversible Markov chains on M. It can also be regarded as a particular case of the theory of generalized (= nonlinear) Dirichlet forms and energy minimizing maps between singular spaces, initiated by Jost (1994) and Korevaar, Schoen (1993) and developed further by Jost (1997a), (1998) and Sturm (1997). We investigate the discrete and continuous time heat flow generated by p and show that various properties of the linear heat flow carry over to this nonlinear heat flow. In particular, we study harmonic maps, i.e. maps which are invariant under the heat flow. These maps are identified with the minimizers of the energy. Received April 2, 2000 / Accepted May 9, 2000 /Published online November 9, 2000     

On holomorphic harmonic morphisms

We study holomorphic harmonic morphisms from K?hler manifolds to almost Hermitian manifolds. When the codomain is also K?hler we get restrictions on such maps in the case of constant holomorphic curvature. We also prove a Bochner-type formula for holomorphic harmonic morphisms which, under certain curvature conditions of the domain, gives insight to the structure of the vertical distribution. We thus prove that when the domain is compact and non-negatively curved, the vertical distribution is totally geodesic. Received: 28 May 2001     

Regularity and uniqueness of p-harmonic maps with small range

We prove the uniqueness of solutions to Dirichlet problem for p-harmonic maps with images in a small geodesic ball of the target manifold. As a consequence, we show that such maps have Hölder continuous derivatives. This gives an extension of a result by Hildebrandt et al. (Acta Math 138:1–16, 1977) concerning harmonic maps.     

On the universal cover of certain exotic Kähler surfaces of negative curvature

We study a class of examples of negatively curved compact Kähler surfaces that are not diffeomorphic to any locally symmetric space. From the analysis of certain totally geodesic curves on these surfaces we deduce that, for infinitely many examples, the natural representation of the fundamental group into PU(2,1) is non-faithful. We also give a new construction of bounded holomorphic functions on the universal cover of our surfaces, based on lifting maps to compact Riemann surfaces.     

Forgetful maps between Deligne–Mostow ball quotients

We study forgetful maps between Deligne–Mostow moduli spaces of weighted points on \mathbbP¹{\mathbb{P}^1} , and classify the forgetful maps that extend to a map of orbifolds between the stable completions. The cases where this happens include the Livné fibrations and the Mostow/Toledo maps between complex hyperbolic surfaces. They also include a retraction of a 3-dimensional ball quotient onto one of its 1-dimensional totally geodesic complex submanifolds.     

Stability and integrability of horizontally conformal maps and harmonic morphisms

In this article, we study stability of minimal fibers and integrability of horizontal distribution for horizontally conformal maps and harmonic morphisms. Let be a horizontally conformal submersion. We prove that if the horizontal distribution is integrable, then any minimal fiber of φ is volume‐stable. This result is an improved version of the main theorem in [15]. As a corollary, we obtain if φ is a submersive harmonic morphism whose fibers are totally geodesic, and the horizontal distribution is integrable, then any fiber of φ is volume‐stable and so such a map φ is energy‐stable if M is compact. We also show that if is a horizontally conformal map from a compact Riemannian manifold M into an orientable Riemannian manifold N which is horizontally homothetic, and if the pull‐back of the volume form of N is harmonic, then the horizontal distribution is integrable and φ is a harmonic morphism.     

On Invariant Submanifolds of C- and S-Manifolds

We characterize totally geodesic invariant submanifolds of C- and S-manifolds. We prove that the second fundamental form is parallel if and only if the submanifold of an S-manifold is totally geodesic. We show that this is not true for the C-manifolds. This revised version was published online in June 2006 with corrections to the Cover Date.     

Semi-invariant Riemannian maps from almost Hermitian manifolds

We construct Gauss–Weingarten-like formulas and define O’Neill’s tensors for Riemannian maps between Riemannian manifolds. By using these new formulas, we obtain necessary and sufficient conditions for Riemannian maps to be totally geodesic. Then we introduce semi-invariant Riemannian maps from almost Hermitian manifolds to Riemannian manifolds, give examples and investigate the geometry of leaves of the distributions defined by such maps. We also obtain necessary and sufficient conditions for semi-invariant maps to be totally geodesic and find decomposition theorems for the total manifold. Finally, we give a classification result for semi-invariant Riemannian maps with totally umbilical fibers.     

Automorphisms and a reduction theorem in a Sasakian space form E2m+1(−3)

Summary We construct definitely the automorphism group of a Sasakian space form ¯M=E ^2m+1 (–3) and study the existence of a totally geodesic invariant submanifold of ¯M tangent to a given invariant subspace in the tangent space of ¯M. We also study the Frenet curves in ¯M under a totally contact geodesic immersion of a contact CR-submanifold into ¯M. The purpose of this paper is to prove a reduction theorem of the codimension for a totally contact geodesic, contact CR-submanifold of ¯M.     

Moduli spaces of vector bundles over a Klein surface

A compact topological surface S, possibly non-orientable and with non-empty boundary, always admits a Klein surface structure (an atlas whose transition maps are dianalytic). Its complex cover is, by definition, a compact Riemann surface M endowed with an anti-holomorphic involution which determines topologically the original surface S. In this paper, we compare dianalytic vector bundles over S and holomorphic vector bundles over M, devoting special attention to the implications that this has for moduli varieties of semistable vector bundles over M. We construct, starting from S, totally real, totally geodesic, Lagrangian submanifolds of moduli varieties of semistable vector bundles of fixed rank and degree over M. This relates the present work to the constructions of Ho and Liu over non-orientable compact surfaces with empty boundary (Ho and Liu in Commun Anal Geom 16(3):617–679, 2008).     

On the geometry of tangent hyperquadric bundles: CR and pseudoharmonic vector fields

We study the natural almost CR structure on the total space of a subbundle of hyperquadrics of the tangent bundle T(M) over a semi-Riemannian manifold (M, g) and show that if the Reeb vector ξ of an almost contact Riemannian manifold is a CR map then the natural almost CR structure on M is strictly pseudoconvex and a posteriori ξ is pseudohermitian. If in addition ξ is geodesic then it is a harmonic vector field. As an other application, we study pseudoharmonic vector fields on a compact strictly pseudoconvex CR manifold M, i.e. unit (with respect to the Webster metric associated with a fixed contact form on M) vector fields X ε H(M) whose horizontal lift X↑ to the canonical circle bundle S¹ → C(M) → M is a critical point of the Dirichlet energy functional associated to the Fefferman metric (a Lorentz metric on C(M)). We show that the Euler–Lagrange equations satisfied by X^↑ project on a nonlinear system of subelliptic PDEs on M. Mathematics Subject Classifications (2000): 53C50, 53C25, 32V20