On Einstein Hermitian manifolds

We show that every compact Einstein Hermitian surface with constant *–scalar curvature is a K?hler surface. In contrast to the 4-dimensional case, it is shown that there exists a compact Einstein Hermitian (4n + 2)-dimensional manifold with constant *–scalar curvature which is not K?hler. This study is supported by Kangwon National University.     

SPACELIKE SUBMANIFOLDS IN THE DE SITTER SPACE Sp^（n＋p）（c）WITH CONSTANT SCALAR CURVATURE

Let M^n be a closed spacelike submanifold isometrically immersed in de Sitter space Sp^(n p)(c)， Denote by R,H and S the normalized scalar curvature,the mean curvature and the square of the length of the second fundamental form of M^n ,respectively. Suppose R is constant and R≤c. The pinching problem on S is studied and a rigidity theorem for M^n immersed in Sp^(n p)(c) with parallel normalized mean curvature vector field is proved. When n≥3, the pinching constant is the best. Thus, the mistake of the paper “Space-like hypersurfaces in de Sitter space with constant scalar curvature”(see Manus Math, 1998,95 :499-505) is corrected. Moreover,the reduction of the codimension when M^n is a complete submanifold in Sp^(n p)(c) with parallel normalized mean curvature vector field is investigated.     

Real Hypersurfaces with Isometric Reeb Flow in Complex Two-Plane Grassmannians

 We classify all real hypersurfaces with isometric Reeb flow in the complex Grassmann manifold G ₂ (ℂ ^m+2 ) of all 2-dimensional linear subspaces in ℂ ^m+2 , m ≥ 3. The second author was supported by Korea Research Foundation. KRF-2001-015-DP0034, Korea. Received April 26, 2001; in revised form December 17, 2001     

Minimal 2-spheres in ℂPn with constant Kähler angle

In 1988 the author and J. Bolton conjectured that a minimally immersed 2-sphere in ℂP ⁿ with constant K?hler angle θ≠ 0, π/2,π necessarily has constant curvature. In 1995 Li Zhen-qi showed that the simplest candidates for counterexamples must be linearly full in ℂP ¹⁰ with tan² (θ/2) = 3/4, and produced an explicit 3-parameter family of them. In the present paper it is shown that these counterexamples may be completely characterised using almost complex curves in the nearly K?hler S ⁶ and that the space of such counterexamples, modulo ambient isometries, is a 14-cell with a single point removed. Received: 7 April 1999     

Kähler manifolds of quasi-constant holomorphic sectional curvature

In correspondence with the manifolds of quasi-constant sectional curvature defined (cf [5], [9]) in the Riemannian context, we introduce in the K?hlerian framework the geometric notion of quasi-constant holomorphic sectional curvature. Some characterizations and properties are given. We obtain necessary and sufficient conditions for these manifolds to be locally symmetric, Ricci or Bochner flat, K?hler η-Einstein or K?hler-Einstein, etc. The characteristic classes are studied at the end and some examples are provided throughout.      

Number of Singularities of Stable Maps

Sharp estimates for the Ricci curvature of a submanifold M ⁿ of an arbitrary Riemannian manifold N ^n+p are established. It is shown that the equality in the lower estimate of the Ricci curvature of M ⁿ in a space form N ^n+p (c) is achieved only when M ⁿ is quasiumbilical with a flat normal bundle. In the case when the codimension p satisfies 1 ≤ p ≤ n − 3, the only submanifolds in N ^n+p (c) on which the Ricci curvature is minimal are the conformally flat ones with a flat normal bundle.      

Stability problems in a theorem of F. Schur

Schur's theorem states that an isotropic Riemannian manifold of dimension greater than two has constant curvature. It is natural to guess that compact almost isotropic Riemannian manifolds of dimension greater than two are close to spaces of almost constant curvature. We take the curvature anisotropy as the discrepancy of the sectional curvatures at a point. The main result of this paper is that Riemannian manifolds in Cheeger's class ℜ(n,d,V,A) withL ₁-small integral anisotropy haveL _p-small change of the sectional curvature over the manifold. We also estimate the deviation of the metric tensor from that of constant curvature in theW _p ² -norm, and prove that compact almost isotropic spaces inherit the differential structure of a space form. These stability results are based on the generalization of Schur' theorem to metric spaces.     

Associated families of pluriharmonic maps¶and isotropy

Like minimal surface immersions in 3-space, pluriharmonic maps into symmetric spaces allow a one-parameter family of isometric deformations rotating the differential (“associated family”); in fact, pluriharmonic maps are characterized by this property. We give a geometric proof of this fact and investigate the “isotropic” case where this family is constant. It turns out that isotropic pluriharmonic maps arise from certain holomorphic maps into flag manifolds. Further, we also consider higher dimensional generalizations of constant mean curvature surfaces which are K?hler submanifolds with parallel (1,1) part of their soecond fundamental form; under certain restrictions there are also characterized by having some kind of (“weak”) associated family. Examples where this family is constant arise from extrinsic K?hler symmetric spaces. Received: 8 July 1997     

The Yamabe problem for almost Hermitian manifolds

The conformal class of a Hermitian metric g on a compact almost complex manifold (M^2m, J) consists entirely of metrics that are Hermitian with respect to J. For each one of these metrics, we may define a J-twisted version of the Ricci curvature, the J-Ricci curvature, and its corresponding trace, the J-scalar curvature s^J. We ask if the conformal class of g carries a metric with constant s^J, an almost Hermitian version of the usual Yamabe problem posed for the scalar curvature s. We answer our question in the affirmative. In fact, we show that (2m−1)s^J−s=2(2m−1)W(ω, ω), where W is the Weyl tensor and ω is the fundamental form of g. Using techniques developed for the solution of the problem for s, we construct an almost Hermitian Yamabe functional and its corresponding conformal invariant. This invariant is bounded from above by a constant that only depends on the dimension of M, and when it is strictly less than the universal bound, the problem has a solution that minimizes the almost complex Yamabe functional. By the relation above, we see that when W (ω, ω) is negative at least one point, or identically zero, our problem has a solution that minimizes the almost Hermitian Yamabe functional, and the universal bound is reached only in the case of the standard 6-sphere equipped with a suitable almost complex structure. When W(ω, ω) is non-negative and not identically zero, we prove that the conformal invariant is strictly less than the universal bound, thus solving the problem for this type of manifolds as well. We discuss some applications.     

Some remarks on Legendrian rectifiable currents

A rectifiable current of dimension n−1 in the sphere bundle Sℝⁿ≃ℝⁿ×S ^{n −1} for euclidean space is Legendrian if it annihilates the contact 1-form α (i.e. T(α∧φ)=0 for all forms φ of degree n−2). Such a current may be naturally associated to any convex set or to any singular real analytic variety, and induces the curvature measures of such a set. We prove that the projection to ℝⁿ of a carrier of a general such T is C ²-rectifiable in the sense of Anzellotti–Serapioni. We deduce that the boundary of a set with positive reach, as well as its singular skeleta, are C ²-rectifiable. In case ∂T= 0 we prove also that the curvature measures associated to T satisfy the analogues of the classical variational formulas for curvature integrals. It follows that such formulas are valid for the curvature measures of subsets of space forms. Received: 3 December 1997/ Revised version: 25 May 1998     

Schouten curvature functions on locally conformally flat Riemannian manifolds

Consider a compact Riemannian manifold (M, g) with metric g and dimension n ≥ 3. The Schouten tensor A _g associated with g is a symmetric (0, 2)-tensor field describing the non-conformally-invariant part of the curvature tensor of g. In this paper, we consider the elementary symmetric functions {σ _k (A _g ), 1 ≤ k ≤ n} of the eigenvalues of A _g with respect to g; we call σ _k (A _g ) the k-th Schouten curvature function. We give an isometric classification for compact locally conformally flat manifolds which satisfy the conditions: A _g is semi-positive definite and σ _k (A _g ) is a nonzero constant for some k ∈ {2, ... , n}. If k = 2, we obtain a classification result under the weaker conditions that σ₂(A _g ) is a non-negative constant and (M ⁿ , g) has nonnegative Ricci curvature. The corresponding result for the case k = 1 is well known. We also give an isometric classification for complete locally conformally flat manifolds with constant scalar curvature and non-negative Ricci curvature. Udo Simon: Partially supported by Chinese-German cooperation projects, DFG PI 158/4-4 and PI 158/4-5, and NSFC.     

Spacelike hypersurfaces with constant scalar curvature

In this paper, we shall give an integral equality by applying the operator □ introduced by S.Y. Cheng and S.T. Yau [7] to compact spacelike hypersurfaces which are immersed in de Sitter spaceS ₁ ⁿ⁺¹ (c) and have constant scalar curvature. By making use of this integral equality, we show that such a hypersurface with constant scalar curvaturen(n−1)r is isometric to a sphere ifr<c. Research partially Supported by a Grant-in-Aid for Scientific Research from the Japanese Ministry of Education, Science and Culture.     

HORIZONTAL DISTRIBUTIONS ON SPACES OF FIBERS

Abstract

Geometric methods for systems of partial differential equations and multiple integral problems in the calculus of variations lead naturally to differentiable manifolds that resemble fiber bundles but do not possess a structure group; in terms of local coordinates, π:B→M_n|(xⁱ, q^α)→(xⁱ), dim(B) = N + n, dim(M_n) = n. The standard notions of horizontal distributions, horizontal and vertical subspaces of T(B), T(B) = V(B) ⊕ H(B), horizontal lifts of curves in M_n, and horizontal and vertical dual subspaces with Λ¹(B) = V*(B) ⊕ H*(B) are shown to be well defined in B. The absence of a structure group is compensated for by an analysis based on the homogeneous ideals V and H that are generated by the canonical bases of V*(B) and H*(B), respectively. The differential system constructed from the generators of the horizontal ideal is shown to lead to a unique system of connection 1-forms and torsion 2-forms under the requirements that they have vacuous intersections with the horizontal ideal. The horizontal ideal is shown to be completely integrable if and only if the torsion 2-forms vanish throughout B, in which case the curvature 2-forms are congruent to zero mod H, and the curvature 2-forms are shown to have a vacuous intersection with H if and only if the horizontal distribution is affine. The paper concludes with a study of the mapping properties of the connection, torsion and curvature. These are significantly more general than those of a fiber bundle since the absence of a structure group allows mappings of the form 'xⁱ = φⁱ(x,q), 'q^α = φ^α (x,q).     

On vertical skew symmetric almost contact 3-structures

We consider a (2m + 3)-dimensional Riemannian manifold M(ξ _r, η^r, g ) endowed with a vertical skew symmetric almost contact 3-structure. Such manifold is foliated by 3-dimensional submanifolds of constant curvature tangent to the vertical distribution and the square of the length of the vertical structure vector field is an isoparametric function. If, in addition, M(ξ _r, η^r, g ) is endowed with an f -structure φ, M, turns out to be a framed f−CR-manifold. The fundamental 2-form Ω associated with φ is a presymplectic form. Locally, M is the Riemannian product of two totally geodesic submanifolds, where is a 2m-dimensional Kaehlerian submanifold and is a 3-dimensional submanifold of constant curvature. If M is not compact, a class of local Hamiltonians of Ω is obtained.     

Curvature bounds for the spectrum of a compact Riemannian manifold of constant scalar curvature

Let (M, g) be an n-dimensional compact and connected Riemannian manifold of constant scalar curvature. If the sectional curvatures of M are bounded below by a constant α > 0, and the Ricci curvature satisfies Ric < (n − 1)αδ, δ ≥ 1, then it is shown that either M is isometric to the n-sphere Sⁿ(α) or else each nonzero eigenvalue λ of the Laplacian acting on the smooth functions of M satisfies the following:

n-dimensional totally real minimal submanifolds of CP n

Let CP ⁿ be the n-dimensional complex projective space with the Study-Fubini metric of constant holomorphic sectional curvature 4 and let M be a compact, orientable, n-dimensional totally real minimal submanifold of CP ⁿ . In this paper we prove the following results.

Optimal pinching constants of odd dimensional homogeneous spaces

In this article we compute the pinching constants of all invariant Riemannian metrics on the Berger space B ¹³=SU(5)/(Sp(2)×_ℤ2S¹) and of all invariant U(2)-biinvariant Riemannian metrics on the Aloff–Wallach space W ⁷ _1,1=SU(3)/S¹ _1,1. We prove that the optimal pinching constants are precisely in both cases. So far B ¹³ and W ⁷ _1,1 were only known to admit Riemannian metrics with pinching constants .?We also investigate the optimal pinching constants for the invariant metrics on the other Aloff–Wallach spaces W ⁷ _k,l =SU(3)/S¹ _k,l . Our computations cover the cone of invariant T²-biinvariant Riemannian metrics. This cone contains all invariant Riemannian metrics unless k/l=1. It turns out that the optimal pinching constants are given by a strictly increasing function in k/l∈[0,1]. Thus all the optimal pinching constants are ≤.?In order to determine the extremal values of the sectional curvature of an invariant Riemannian metric on W ⁷ _k,l we employ a systematic technique, which can be applied to other spaces as well. The computation of the pinching constants for B ¹³ is reduced to the curvature computation for two proper totally geodesic submanifolds. One of them is diffeomorphic to ℂℙ³/ℤ₂ and inherits an Sp(2)-invariant Riemannian metric, and the other is W ⁷ _1,1 embedded as recently found by Taimanov. This approach explains in particular the coincidence of the optimal pinching constants for W ⁷ _1,1 and the Berger space B ¹³. Oblatum 9-XI-1998 & 3-VI-1999 / Published online: 20 August 1999     

Asymptotic behaviour and the moduli space of doubly-periodic instantons

We study doubly-periodic instantons, i.e. instantons on the product of a 1-dimensional complex torus T with a complex line ℂ, with quadratic curvature decay. We determine the asymptotic behaviour of these instantons, constructing new asymptotic invariants. We show that the underlying holomorphic bundle extends to T×ℙ¹. The converse statement is also true, namely a holomorphic bundle on T×ℙ¹ which is flat on the torus at infinity, and satisfies a stability condition, comes from a doubly-periodic instanton. Finally, we study the hyperk?hler geometry of the moduli space of doubly-periodic instantons, and prove that the Nahm transform previously defined by the second author is a hyperk?hler isometry with the moduli space of certain meromorphic Higgs bundles on the dual torus. Received June 8, 2000 / final version received February 1, 2001?Published online April 3, 2001     

A class of Riemannian manifolds that pinch when evolved by Ricci flow

The purpose of this paper is to construct a set of Riemannian metrics on a manifold X with the property that will develop a pinching singularity in finite time when evolved by Ricci flow. More specifically, let , where N ⁿ is an arbitrary closed manifold of dimension n≥ 2 which admits an Einstein metric of positive curvature. We construct a (non-empty) set of warped product metrics on the non-compact manifold X such that if , then a smooth solution , t∈[0,T) to the Ricci flow equation exists for some maximal constant T, 0<T<∞, with initial value , and