On the curvature of the diffeomorphisms group

An expression for the sectional curvature ofSDIFF(M) (the group of diffeomorphism preserving Riemannian density on a closed manifoldM) is obtained. In the case of a locally Euclidean manifoldM, the negativeness of curvature that implies the instability of solutions of Euler equations of ideal incompressible fluids onM is established.     

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 .     

Strong p-completeness of stochastic differential equations and the existence of smooth flows on noncompact manifolds

Summary Here we discuss the regularity of solutions of SDE's and obtain conditions under which a SDE on a complete Riemannian manifoldM has a global smooth solution flow, in particular improving the usual global Lipschitz hypothesis whenM=R ⁿ . There are also results on non-explosion of diffusions.Research supported by SERC grant GR/H67263     

Riemannian manifolds carrying a pair of skew symmetric conformal vector fields

We deal with a Riemannian manifoldM carrying a pair of skew symmetric conformal vector fields (X, Y). The existence of such a pairing is determined by an exterior differential system in involution (in the sense of Cartan). In this case,M is foliated by 3-dimensional totally geodesic submanifolds. Additional geometric properties are proved. Supported by a JSPS postdoctoral fellowship.     

A sharpL q-Liouville theorem forp-harmonic functions

We studyL ^q-Liouville properties of nonnegativep-superharmonic and, respectively,p-subharmonic functions on a complete Riemannian manifoldM. In particular, we prove that everyp-harmonic functionu ∈L ^q (M) is constant ifq>p−1. Supported by the Academy of Finland, Project 6355.     

The center foliation of an affine diffeomorphism

Given an affine (i.e. connection-preserving) diffeomorphismf of a Riemannian manifoldM, we consider itscenter foliation, N, comprised by the directions that neither expand nor contract exponentially under the action generated byf. The main remarks made here (Corollary 3 and Theorem 7) are: There exists a metric compatible with the Levi-Civita connection for which the universal cover ofM decomposes isometrically as the Riemannian product of the universal cover of a leaf ofN (these covers are all isometric) and the Euclidean space; and ifN is one-dimensional,M is flat and the foliation is (up to finite cover) the fiber foliation of a Riemannian submersion onto a flat torus.     

TheL2-metric on the moduli space ofSU(2)-instantons with instanton number 1 over the Euclidean 4-space

TheL ²-metric {ie311-1} on the moduli spaceM ₁(Q) of self-dualSU(2)-connections with instanton number 1 over the Euclidean 4-space is described. It is shown that the Riemannian manifold (M ₁(Q), {ie311-2}) is isometric to R⁺ × R⁴ with the Euclidean metric.Supported by Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 288     

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.     

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 .     

Rigidity theorem for harmonic maps from Riemannian manifold to Grassmannian manifold

We first study the Grassmannian manifoldG _n (R^n+p)as a submanifold in Euclidean space ⁿ (R ^n+p). Then we give a local expression for each map from Riemannian manifoldM toG ⁿ (R ^n+p) ⁿ (R ^n+p), and use the local expression to establish a formula which is satisfied by any harmonic map fromM toG _n (R ^n+p). As a consequence of this formula we get a rigidity theorem.     

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     

Bubbling phenomena of Palais-Smale-like sequences ofm-harmonic type systems

In this paper, we study the bubbling phenomena of weak solution sequences of a class of degenerate quasilinear elliptic systems ofm-harmonic type. We prove that, under appropriate conditions, the energy is preserved during the bubbling process. The results apply tom-harmonic maps from a closed Riemannian manifoldM to a Riemannian homogeneous space, and tom-harmonic maps with constant volumes, and also to certain Palais-Smale sequences.     

Minimal annuli in Riemannian manifolds

In this paper, a multiple solution theorem for minimal annuli coboundaries in a Riemannian manifoldN is established. Especilly, when the target manifoldN is the standard sphereS ⁿ , it implies the existence of at least two minimal annuli with given pair of wires (Γ¹, Γ²) as their common boundaries θ. This paper was completed when the author was a postdoctoral student under the guidance of Prof. C.K. Peng. Many thanks are due to him for his encouragement and much help.     

Tubes and eigenvalues for negatively curved manifolds

We investigate the structure of the spectrum near zero for the Laplace operator on a complete negatively curved Riemannian manifoldM. If the manifold is compact and its sectional curvaturesK satisfy 1 ≤K < 0, we show that the smallest positive eigenvalue of the Laplacian is bounded below by a constant depending only on the volume ofM. Our result for a complete manifold of finite volume with sectional curvatures pinched between −a² and −1 asserts that the number of eigenvalues of the Laplacian between 0 and (n− 1)²/4 is bounded by a constant multiple of the volume of the manifold with the constant depending ona and the dimension only. Research supported in part by the Swiss National Science Foundation, the US National Science Foundation, and the PSC-CUNY Research Award Program.     

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     

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 torsion of a submanifold of a Riemannian manifold

For a submanifoldM ⁿ of a Riemannian manifoldM ^q, the concept of a torsion bivector at the point x M ⁿ for given one- and two-dimensional directions fromT _x M ⁿ is introduced using only the first and second fundamental forms ofM ⁿ. Its relation to the concept of Gaussian torsion is then established. It is proved that: 1) equality to zero of the torsion bivector is necessary and, whenM ⁿ is a nondevelopable surface of a space of constant curvature with nonzero second fundamental form, is also sufficient for the "flattening" ofM ⁿ into some totally geodesicM ⁿ⁺¹ inM ^q; 2) when n = 2, the independence of the nonzero torsion bivector of direction characterizes a minimalM ² inM ^q.Translated from Ukrainskii Geometricheskii Sbornik, No. 34, pp. 39–42, 1991.     

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.     

Some remarks on the (f, g)-linear connections

In this note we consider structures defined on a differentiable manifoldM by a tensor fieldf of type (1,1) satisfying the conditionf ⁴+f ²=0 and by a Riemannian structureg such thatg(f(X),Y)=−g(Xf(Y)) for all vector fieldsX,YεT ¹ ₀(M). Then we determine linear connections compatible with those structures. Facultatea de Matematica, Str. Academiei 14, 70109 Bucaresti, Romania. Published in Lietuvos Matematikos Rinkinys, Vol. 37, No. 3, pp. 383–387, July–September, 1997.     

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).