Conformal deformations of integral pinched 3-manifolds

In this paper we prove that, under an explicit integral pinching assumption between the L²-norm of the Ricci curvature and the L²-norm of the scalar curvature, a closed 3-manifold with positive scalar curvature admits a conformal metric of positive Ricci curvature. In particular, using a result of Hamilton, this implies that the manifold is diffeomorphic to a quotient of S³. The proof of the main result of the paper is based on ideas developed in an article by M. Gursky and J. Viaclovsky.     

Torsion and homogeneity on contact metric three-manifolds

We show that a three-dimensional contact metric manifold is locally homogeneous if and only if it is ball-homogeneous and satisfies the condition ∇_ξτ=2aτϕ, with a constant. Then, we relate the condition ∇_ξτ=0 with the existence of taut contact circles on a compact three-dimensional contact metric manifold. Entrata in Redazione il 20 gennaio 1999. Supported by funds of the University of Lecce and the M.U.R.S.T. Work made within the program of G.N.S.A.G.A.-C.N.R.     

Isotropic ppmc immersions

Isometric immersions with parallel pluri-mean curvature (“ppmc”) in euclidean n-space generalize constant mean curvature (“cmc”) surfaces to higher dimensional Kähler submanifolds. Like cmc surfaces they allow a one-parameter family of isometric deformations rotating the second fundamental form at each point. If these deformations are trivial the ppmc immersions are called isotropic. Our main result drastically restricts the intrinsic geometry of such a submanifold: Locally, it must be a symmetric space or a Riemannian product unless the immersion is holomorphic or a superminimal surface in a sphere. We can give a precise classification if the codimension is less than 7. The main idea of the proof is to show that the tangent holonomy is restricted and to apply the Berger-Simons holonomy theorem.     

Rigidity and sphere theorem for manifolds with positive Ricci curvature

LetM be a complete Riemannian manifold with Ricci curvature having a positive lower bound. In this paper, we prove some rigidity theorems forM by the existence of a nice minimal hypersurface and a sphere theorem aboutM. We also generalize a Myers theorem stating that there is no closed immersed minimal submanifolds in an open hemisphere to the case that the ambient space is a complete Riemannian manifold withk-th Ricci curvature having a positive lower bound. Supported by the JSPS postdoctoral fellowship and NSF of China     

A Bochner formula for almost-quaternionic-Hermitian structures

An integral formula is derived, relating the six irreducible components of the intrinsic torsion of an Sp_nSp₁ structure on a compact 4n-dimensional manifold with the Riemann curvature tensor. Some consequences of the formula are studied.     

Ricci flat metrics with bidimensional null orbits and non-integrable orthogonal distribution

We study Ricci flat 4-metrics of any signature under the assumption that they allow a Lie algebra of Killing fields with 2-dimensional orbits along which the metric degenerates and whose orthogonal distribution is not integrable. It turns out that locally there is a unique (up to a sign) metric which satisfies the conditions. This metric is of signature (++−−) and, moreover, homogeneous possessing a 6-dimensional symmetry algebra.     

Light-like CR hypersurfaces of indefinite Kähler manifolds

We study a new class of real hypersurfaces called Light-like CR hypersurfaces, of indefinite Kahler manifolds, and claim several new results of geometrical/physical significance. In particular, we show that our study has a direct relation with the physically important asymptotically flat spacetimes; which further lead to the Twistor theory of Penrose and the Heaven theory of Newman. As the induced connection, on the degenerate hypersurface, may not be a metric connection, we overcome this difficulty by using differential geometric technique and deduce the embedding conditions called Gauss-Codazzi equations. Finally, we find the integrability conditions for all the possible distributions and specialize the embedding conditions when the ambient space is a complex space form. We add to the list of totally umbilical nondegenerate hypersurfaces [16] the totally umbilical light-like cone, in the degenerate case, and prove the nonexistence of totally umbilical light-like CR hypersurfaces in ¯M(c) withc 0 (see Yano and Kon [22] and Tashiro and Tachibana [20] for the nondegenerate case).     

Pseudo-Riemannian 3-manifolds with prescribed distinct constant Ricci eigenvalues

We study three-dimensional pseudo-Riemannian manifolds having distinct constant principal Ricci curvatures. These spaces are described via a system of differential equations, and a simple characterization is proved to hold for the locally homogeneous ones. We then generalize the technique used in [O. Kowalski, F. Prüfer, On Riemannian 3-manifolds with distinct constant Ricci eigenvalues, Math. Ann. 300 (1994) 17-28] for Riemannian manifolds and construct explicitly homogeneous and non-homogeneous pseudo-Riemannian metrics in R³, having the prescribed principal Ricci curvatures.     

Classification of pinched positive scalar curvature manifolds

Let (Mn,g), n?3, be a smooth closed Riemannian manifold with positive scalar curvature Rg. There exists a positive constant C=C(M,g) defined by mean curvature of Euclidean isometric immersions, which is a geometric invariant, such that Rg?n(n−1)C. In this paper we prove that Rg=n(n−1)C if and only if (Mn,g) is isometric to the Euclidean sphere Sn(C) with constant sectional curvature C. Also, there exists a Riemannian metric g on Mn such that the scalar curvature satisfies the pinched condition

     

Special directions on contact metric three-manifolds

Blair [5] has introduced special directions on a contact metric 3-manifolds with negative sectional curvature for plane sections containing the characteristic vector field and, when is Anosov, compared such directions with the Anosov directions. In this paper we introduce the notion of Anosov-like special directions on a contact metric 3-manifold. Such directions exist, on contact metric manifolds with negative -Ricci curvature, if and only if the torsion is -parallel, namely (1.1) is satisfied. If a contact metric 3-manifold M admits Anosov-like special directions, and is -parallel, where is the Berger-Ebin operator, then is Anosov and the universal covering of M is the Lie group (2,R). We note that the notion of Anosov-like special directions is related to that of conformally Anosow flow introduced in [9] and [14] (see [6]).Supported by funds of the M.U.R.S.T. and of the University of Lecce. 1991.     

Maximal surfaces in a certain 3-dimensional homogeneous spacetime

A generalized integral representation formula for spacelike maximal surfaces in a certain 3-dimensional homogeneous spacetime is obtained. This spacetime has a solvable Lie group structure with left invariant metric. The normal Gauß map of maximal surfaces in the homogeneous spacetime is discussed and the harmonicity of the normal Gauß map is studied.     

Second variation and Legendrian stabilities of minimal Legendrian submanifolds in Sasakian manifolds

First, we derive a new second variation formula which holds for minimal Legendrian submanifolds in Sasakian manifolds. Using this, we prove that any minimal Legendrian submanifold in an η-Einstein Sasakian manifold with “nonpositive” η-Ricci constant is stable. Next we introduce the notion of the Legendrian stability of minimal Legendrian submanifolds in Sasakian manifolds. Using our second variation formula, we find a general criterion for the Legendrian stability of minimal Legendrian submanifolds in η-Einstein Sasakian manifolds with “positive” η-Ricci constant.     

Minimal 2-spheres in a complex projective space

In this paper conformal minimal 2-spheres immersed in a complex projective space are studied by applying Lie theory and moving frames. We give differential equations of Kähler angle and square length of the second fundamental form. By applying these differential equations we give characteristics of conformal minimal 2-spheres of constant Kähler angle and obtain pinching theorems for curvature. We also discuss conformal minimal 2-spheres of constant normal curvature and prove that there does not exist any linearly full minimal 2-sphere immersed in a complex projective space CPn (n>2) with non-positive constant normal curvature. We also prove that a linearly full minimal 2-sphere immersed in a complex projective space CPn (n>2) with constant normal curvature and constant Kähler angle is of constant curvature.     

Some Classes of Almost Anti-Hermitian Structures on the Tangent Bundle

In [11] we have considered a family of natural almost anti-Hermitian structures (G, J) on the tangent bundle TM of a Riemannian manifold (M, g), where the semi-Riemannian metric G is a lift of natural type of g to TM, such that the vertical and horizontal distributions VTM, HTM are maximally isotropic and the almost complex structure J is a usual natural lift of g of diagonal type interchanging VTM, HTM (see [5], [15]). We have obtained the conditions under which this almost anti-Hermitian structure belongs to one of the eight classes of anti-Hermitian manifolds obtained in the classification given in [1]. In this paper we consider another semi-Riemannian metric G on TM such that the vertical and horizontal distributions are orthogonal to each other. We study the conditions under which the above almost complex structure J defines, together with G, an almost anti-Hermitian structure on TM. Next, we obtain the conditions under which this structure belongs to one of the eight classes of anti-Hermitian manifolds obtained in the classification in [1].Partially supported by the Grant 100/2003, MECT-CNCSIS, România.     

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.     

New examples of weakly symmetric spaces

We provide some new examples of weakly symmetric spaces inside the class of complete, simply connected Riemannian manifolds equipped with a complete unit Killing vector field such that the reflections with respect to its flow lines are global isometries.Supported by the Consejería de Educación del Gobierno de Canarias     

Extrinsic homogeneity of parallel submanifolds II

We discuss the question whether a (complete) parallel submanifold M of a Riemannian symmetric space N is an (extrinsically) homogeneous submanifold, i.e. whether there exists a subgroup of the isometries of N which acts transitively on M. In a previous paper, we have discussed this question in case the universal covering space of M is irreducible. It is the subject of this paper to generalize this result to the case when the universal covering space of M has no Euclidian factor.     

On complete noncompact submanifolds with constant mean curvature and finite total curvature in Euclidean spaces

In this paper, we consider a complete noncompact n-submanifold M with parallel mean curvature vector h in an Euclidean space. If M has finite total curvature, we prove that M must be minimal, so that M is an affine n-plane if it is strongly stable. This is a generalization of the result on strongly stable complete hypersurfaces with constant mean curvature in Received: 30 June 2005     

The mapping properties of filling radius and packing radius and their applications

Let be a map between closed, oriented Riemannian n-manifolds. It is shown that FillRad(W)?dil(f)⋅FillRad(V), if |deg(f)|=1. By this mapping property, we obtain an estimate from below for the filling radius of a closed, oriented, nonpositively curved manifold, or a manifold with sectional curvature bounded above by a positive constant. In addition, a similar mapping property of packing radius and a corollary are also obtained.