Estimates for the curvature of a submanifold which lies inside a tube

We give some estimates for the supremun of the sectional curvature of a submanifold N of a Riemannian or Kaehlerian manifoldM such that N is contained in a tube about a submanifoldP ofM. These estimates depend on the sectional curvature ofM, the Weingarten map ofP and the radius of the tube. Then we apply them to get theorems of non immersibility.Work partially supported by a DGICYT NO. PB94-0972     

Foliations with leaves of nonpositive curvature

We answer a question of Gromov ([G2]) in the codimension 1 case: ifF is a codimension 1 foliation of a compact manifoldM with leaves of negative curvature, thenπ ₁(M) has exponential growth. We also prove a result analogous to Zimmer’s ([Z2]): ifF is a codimension 1 foliation on a compact manifold with leaves of nonpositive curvature, and ifπ ₁(M) has subexponential growth, then almost every leaf is flat. We give a foliated version of the Hopf theorem on surfaces without conjugate points. Partially supported by NSF Grant #DMS 9403870.     

Manifolds whose curvature operator has constant eigenvalues at the basepoint

Osserman conjectured that if the curvature operatorR of a Riemannian manifoldM has constant eigenvalues, thenM is locally a rank-1 symmetric space or is flat. The pointwise question is considerably more complicated. We present examples of Riemannian manifolds so thatR has constant eigenvalues at the basepoint, butR is not the curvature operator of a rank-1 symmetric space. Research partially supported by the NSF and IHES.     

Harmonic maps from a Riemannian manifold with a pole into an Hadamard manifold with negative sectional curvatures

In this paper we show a nonexistence result for harmonic maps with a rotational nondegeneracy condition from a Riemannian manifoldM with polep ₀ to a negatively curved Hadamard manifold under the condition that the metric tensor ofM is bounded and that the sectional curvature ofM at a pointp is bounded from below by −c dist(p ₀,p)⁻² (c: a positive constant) as dist(p ₀,p)→∞. Partly supported by Grants-in-Aid for Scientific Research, The Ministry of Education, Science and Culture, Japan     

Upper bounds of small Eigenvalues of the Dirac operator and isometric immersions

We show that a topologically determined number of eigenvalues of the Dirac operatorD of a closed Riemannian spin manifoldM of even dimensionn can be bounded by the data of an isometric immersion ofM into the Euclidian spaceR ^N . From this we obtain similar bounds of the eigenvalues ofD in terms of the scalar curvature ofM ifM admits a minimal immersion intoS ^N or,ifM is complex, a holomorphic isometric immersion intoPC ^N .     

A proof of the Chern-Lashof conjecture in dimensions greater than five

Summary Chern and Lashof ([1], [2]) conjectured that if a smooth manifoldM ^m has an immersion intoR ^w, then the best possible lower bound for its total absolute curvature is the Morse number μ(M). We give a proof of this whenm>5. Under the same dimension restriction, our methods allow us to show that μ(M) is still the best possible lower bound among immersions within a fixed regular homotopy class except in the casew=m+1=even, for which the best lower bound is max {μ(M), 2 |d|}, whered the degree of the Gauss map. This work was supported in part by NSERC grant A4621. In addition the author would like to thank the University of Geneva, the Swiss Science Foundation and the University of Paris VII for their hospitality while parts of this paper were written.     

On the spectral synthesis property and its application to partial differential equations

LetM be a (n−1)-dimensional manifold inR ⁿ with non-vanishing Gaussian curvature. Using an estimate established in the early work of the author [4], we will improve the known result of Y. Domar on the weak spectral synthesis property by reducing the smoothness assumption upon the manifoldM. Also as an application of the method, a uniqueness property for partial differential equations with constant coefficients will be proved, which for some specific cases recovers or improves H?rmander's general result.     

Projective homogeneity

We consider aC simply connected manifoldM endowed with a projective structureP and under an additional hypothesis on the projective curvature tensor, we find necessary and sufficient conditions in order thatM turns out to be a reductive homogeneous spaceG/H whereG is a Lie group acting onM as a group of automorphisms ofP.     

Stationary harmonic maps into complete Riemannian manifold

The stationary for harmonic maps is considered from a Riemannian manifoldM into a complete Riemannian manifoldN without boundary, and it is proved that its singular set is contained inQ ¹ 2MQ ³ Project supported partially by the Development Foundation Science of Shanghai, China.     

Totally geodesic hypersurfaces in manifolds of nonpositive curvature

In this paper we determine the structure of an embedded totally geodesic hypersurfaceF or, more generally, of a totally geodesic hypersurfaceF without selfintersections under arbitrarily small angles in a compact manifoldM of nonpositive sectional curvature. Roughly speaking, in the case of locally irreducibleM the result says thatF has only finitely many ends, and each end splits isometrically asK×(0, ∞), whereK is compact. This article was processed by the author using the Springer-Verlag T_EX PJour1g macro package 1991.     

On the solutions to certain laplace inequalities with applications to geometry of submanifolds

LetM be a complete Riemannian manifold with Ricci curvature bounded from below. We give an explicit estimate for the size of the negative sets of solutions to the differential inequality Δu ≥λu where Δ is the Laplacian and λ is a negative constant. This inequality arises naturally when we study the lengthH of the mean curvature of an isometric immersionf of M into another Riemannian manifoldN with curvature bounded above by some constantκ. Suppose that the image f(M) does not meet the cut locus of some pointo ∈ N. As a consequence of our estimate, we prove that, givenρ > 0, if supH is less than a certain explicit expression μ(κ, ρ) inρ andκ on any domainU that contains an inscribed ball of radius greater than an explicitly computable numberR, then the diameter of the setf(U) inN must exceed 2ρ. Moreover, if supH = μ(κ, ρ) onM and the diameter off(M) inN equals 2ρ, thenf is a minimal immersion into a distance sphere of radiusρ inN.     

Ehresmann connection for the canonical foliation on a manifold over a local algebra

For a canonical foliation on a manifoldM ^A over a local algebra, theA-affine horizontal distribution complementary to the leaves, similar to the horizontal distribution of a higher order connection on the fiber bundle ofA-jets, is defined. In the case of a complete manifoldM ^A, theA-affine horizontal distribution is proved to be an Ehresmann connection in the sense of Blumental-Hebda. It is shown that theA-affine horizontal distribution onM ^A exists if and only if the Atiyah class of a certain foliated principal bundle vanishes.Translated fromMatematicheskie Zametki, Vol. 59, No. 2, pp. 303–310, February, 1996.     

The canonical spectral sequences for poisson manifolds

For a compact symplectic manifoldM of dimension 2n, Brylinski proved that the canonical homology groupH _k ^can (M) is isomorphic to the de Rham cohomology groupH ^2n-k (M), and the first spectral sequence {E ^r (M)} degenerates atE ¹(M). In this paper, we show that these isomorphisms do not exist for an arbitrary Poisson manifold. More precisely, we exhibit an example of a five-dimensional compact Poisson manifoldM ⁵ for whichH ₁ ^can (M ⁵) is not isomorphic toH ⁴(M ⁵), andE ¹(M ⁵) is not isomorphic toE ²(M ⁵). This work has been partially supported through grants DGICYT (Spain), Projects PB91-0142 and PB89-0571, PB94-0633-C02-02; and through grants UPV, Project 127.310-EA 191/94, 127.310-EC248/96.     

Standard pseudo-Hermitian structure on manifolds and seifert fibration

A strictly pseudoconvex pseudo-Hermitian manifoldM admits a canonical Lorentz metric as well as a canonical Riemannian metric. Using these metrics, we can define a curvaturelike function onM. AsM supports a contact form, there exists a characteristic vector field dual to the contact structure. If induces a local one-parameter group ofCR transformations, then a strictly pseudoconvex pseudo-Hermitian manifoldM is said to be a standard pseudo-Hermitian manifold. We study topological and geometric properties of standard pseudo-Hermitian manifolds of positive curvature or of nonpositive curvature . By the definition, standard pseudo-Hermitian manifolds are calledK-contact manifolds by Sasaki. In particular, standard pseudo-Hermitian manifolds of constant curvature turn out to be Sasakian space forms. It is well known that a conformally flat manifold contains a class of Riemannian manifolds of constant curvature. A sphericalCR manifold is aCR manifold whose Chern-Moser curvature form vanishes (equivalently, Weyl pseudo-conformal curvature tensor vanishes). In contrast, it is emphasized that a sphericalCR manifold contains a class of standard pseudo-Hermitian manifolds of constant curvature (i.e., Sasakian space forms). We shall classify those compact Sasakian space forms. When 0, standard pseudo-Hermitian closed aspherical manifolds are shown to be Seifert fiber spaces. We consider a deformation of standard pseudo-Hermitian structure preserving a sphericalCR structure.Dedicated to Professor Sasao Seiya for his sixtieth birthday     

On the canonical extension of a differentiable manifold

In this paper, the differential geometry of second canonical extension² M of a differentiable manifoldM is studied. Some vector fields tangent to² M inTTM are determined. In addition we obtain that the second canonical extensions ofM and a totally geodesic submanifold inM are totally geodesic submanifolds inTTM and² M respectively.     

On the normal bundle of a complex submanifold of a locally conformal Kaehler manifold

We investigate the existence of parallel sections in the normal bundle of a complex submanifold of a locally conformal Kaehler manifold with positive holomorphic bisectional curvature. Also, ifM is a quasi-Einstein generalized Hopf manifold then we show that any complex submanifoldM with a flat normal connection ofM is quasi-Einstein, too, provided thatM is tangent to the Lee field ofM. As an application of our results we study the geometry of the second fundamental form of a complex submanifold in the locally conformal Kaehler sphereQ _m (of a complex Hopf manifoldS ^2m+1 ×S ¹).     

The gradient of certain harmonic functions on manifolds of almost nonnegative Ricci curvature

In this paper we prove that when the Ricci curvature of a Riemannian manifoldM ⁿ is almost nonnegative, and a ballB _L (p)⊂M ⁿ is close in Gromov-Hausdorff distance to a Euclidean ball, then the gradient of the harmonic functionb defined in [ChCo1] does not vanish. In particular, these functions can serve as harmonic coordinates on balls sufficiently close to an Euclidean ball. The proof, is based on a monotonicity theorem that generalizes monotonicity of the frequency for harmonic functions onR ⁿ .     

Conformal deformations of prescribing scalar curvature on Riemannian manifolds with negative curvature

We study the conformal deformation for prescribing scalar curvature function on Cartan-Hadamard manifoldM ⁿ (n≥3) with strongly negative curvature. By employing the supersubsolution method and a careful construction for the supersolution, we obtain the best possible asymptotic behavior for near infinity so that the problem of complete conformal deformation is solvable. In more general cases, we prove an asymptotic estimation on the solutions of the conformal scalar curvature equation. Project partially supported by the NNSF of China     

Generating closed 2-cell embeddings in the torus and the projective plane

If a graphG is embedded in a manifoldM such that all faces are cells bounded by simple closed curves we say that this is a closed 2-cell embedding ofG inM. We show how to generate the 2-cell embeddings in the projective plane from two minimal graphs and the 2-cell embeddings in the torus from six minimal graphs by vertex splitting and face splitting.     

Zeta-functions for expanding maps and Anosov flows

Given a real-analytic expanding endomorphism of a compact manifoldM, a meromorphic zeta function is defined on the complex-valued real-analytic functions onM. A zeta function for Anosov flows is shown to be meromorphic if the flow and its stable-unstable foliations are real-analytic.