On warped product immersions

Warped product immersions appeared naturally in several recent studies. In this article we study fundamental geometric properties of such immersions. In addition we prove that every non-flat complex space form of complex dimension greater than one does not admit a warped product representation. As an application we obtain an improvement of an earlier result in [3] concerning warped products in real space forms.     

Geometry of warped product pseudo-slant submanifolds of Kenmotsu manifolds

In this paper, we study pseudo-slant submanifolds and their warped products in Kenmotsu manifolds. We obtain the necessary conditions that a pseudoslant submanifold is locally a warped product and establish an inequality for the squared norm of the second fundamental form in terms of the warping function. The equality case is also considered.     

Harmonic morphisms of warped product type from Einstein manifolds

Weitzenb?ck type identities for harmonic morphisms of warped product type are developed which lead to some necessary conditions for their existence. These necessary conditions are further studied to obtain many nonexistence results for harmonic morphisms of warped product type from Einstein manifolds. Received: 14 March 2006     

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

     

On the fundamental group of some open manifolds

We study fundamental groups of noncompact Riemannian manifolds. We find conditions which ensure that the fundamental group is trivial, finite or finitely generated.     

Collapse of warped foliations

The aim of this note is to generalize the concept of warped product to a foliated manifold (M,F,g) as follows: If is a smooth function constant along the leaves of the foliation F then new metric structure gf on the manifold M is constructed as follows: gf(v,w)=f²g(v,w) if v,w are tangent to F and gf(v,w)=g(v,w) if v or w is perpendicular to F. A foliated manifold (M,F,gf) is called warped foliation while f is called warping function.Next, if is a sequence of warping functions on M, the question of the existence of the limit in Gromov-Hausdorff of a sequence ((M,F,gfn))_n∈N warped foliation is asked. A number of examples is considered such foliations with dense leaf or foliations consisting of finite number of Reeb components. Next, sufficient and necessary condition of converging in Gromov-Hausdorff sense of a Riemannian foliation with all leaves compact to the space of leaves with a metric defined by Hausdorff distance of leaves is developed. Finally some results on Hausdorff foliations with all leaves compact are shown.     

Four-dimensional Einstein-like manifolds and curvature homogeneity

The aim of this paper is to classify 4-dimensional Einstein-like manifolds whose Ricci tensor has constant eigenvalues (this being a special kind of curvature homogeneity condition). We give a full classification when the Ricci tensor is of Codazzi type; when the Ricci tensor is cyclic parallel, we classify all such manifolds when not all Ricci curvatures are distinct. In this second case we find a one-parameter family of Riemannian metrics on a Lie groupG as the only possible ones which are irreducible and non-symmetric.     

Complete hypersurfaces immersed in a semi-Riemannian warped product

The aim of this paper is to study the uniqueness of complete hypersurfaces immersed in a semi-Riemannian warped product whose warping function has convex logarithm and such that its fiber has constant sectional curvature. By using as main analytical tool a suitable maximum principle for complete noncompact Riemannian manifolds and supposing a natural comparison inequality between the r-th mean curvatures of the hypersurface and that ones of the slices of the region where the hypersurface is contained, we are able to prove that a such hypersurface must be, in fact, a slice.     

On rigidity of hypersurfaces with constant curvature functions in warped product manifolds

In this paper, we first investigate several rigidity problems for hypersurfaces in the warped product manifolds with constant linear combinations of higher order mean curvatures as well as “weighted” mean curvatures, which extend the work (Brendle in Publ Math Inst Hautes Études Sci 117:247–269, 2013; Brendle and Eichmair in J Differ Geom 94(94):387–407, 2013; Montiel in Indiana Univ Math J 48:711–748, 1999) considering constant mean curvature functions. Secondly, we obtain the rigidity results for hypersurfaces in the space forms with constant linear combinations of intrinsic Gauss–Bonnet curvatures $L_k$ . To achieve this, we develop some new kind of Newton–Maclaurin type inequalities on $L_k$ which may have independent interest.     

On the rigidity of constant mean curvature complete vertical graphs in warped products

We investigate constant mean curvature complete vertical graphs in a warped product, which is supposed to satisfy an appropriated convergence condition. In this setting, under suitable restrictions on the values of the mean curvature and the norm of the gradient of the height function, we obtain rigidity theorems concerning to such graphs. Furthermore, applications to the hyperbolic and Euclidean spaces are given.     

Unrectifiability and rigidity in stratified groups

In the class of stratified groups endowed with a left invariant Carnot-Carathéodory distance, we give an algebraic characterization of purely unrectifiable groups and we study rigidity properties. The main feature of our approach is the use of a suitable area formula with respect to the Carnot-Carathéodory distance.Received: 8 March 2004     

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.      

Hermitian metric rigidity on compact foliated manifolds

We prove a Hermitian metric rigidity theorem for leafwise symmetric Kaehler metrics on compact manifolds with smooth foliations. This provides applications to the study of the geometry of foliations as well as Kaehler manifolds that contain some symmetric geometry.     

A rigidity theorem for complete CMC hypersurfaces in Lorentz manifolds

In this paper we use the standard formula for the Laplacian of the squared norm of the second fundamental form and the asymptotic maximum principle of H. Omori and S.T. Yau to classify complete CMC spacelike hypersurfaces of a Lorentz ambient space of nonnegative constant sectional curvature, under appropriate bounds on the scalar curvature.     

Doubly warped product <Emphasis Type="Italic">CR</Emphasis>-submanifolds in locally conformal Kähler manifolds

In this paper we study doubly warped product CR submanifolds in locally conformal K?hler manifolds, and we found a B.Y. Chen’s type inequality for the second fundamental form of these submanifolds. Beneficiary of a CNR-NATO Advanced Research Fellowship pos. 216.2167 Prot. n. 0015506.     

Volume growth of some uniformly contractible Riemannian manifolds

It is shown that if a uniformly contractible Riemannian n-manifold (M,g) is K-quasi-isometric to an n-dimensional normed space\((V^{n},\|\cdot\|)\), (K ≥  1), then\(\liminf_{R\rightarrow \infty}\frac{{Vol}_g( {Ball}_{R})}{R^{n}\omega_{n}}\geq\frac{1}{K^{2n}}\) where ω _n is the volume of the unit Euclidean ball. In particular, if M is uniformly contractible and\(d_{GH}((M,d_g), (V^n,\|\cdot\|)) < \infty \), then M has at least Euclidean volume growth. This corollary covers an earlier result by Burago and Ivanov. Our results are motivated by a volume growth theorem contained in Gromov’s book [Gromov in Progress in Mathematics, vol. 152, Birkhäuser, Boston, 1999, p. 256], we give a detailed proof of this theorem. Using the same argument, we also derive a generalization of the theorem which is pointed out by Gromov.     

On an existence theorem of Wagner manifolds

In their common paper [An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.) 43 (1997) 307-321] the authors give a condition for a 1-form β to be the perturbation of a Riemannian manifold (M, α) such that the manifold equipped with any (α, β) -metric is a Wagner manifold with respect to the Wagner connection induced by β. The condition shows that its covariant derivative with respect to the Lévi-Civita connection must be of a special form

(∇β)(X,Y)=∥β^#∥2α(X,Y)−β(X)β(Y)     

On the volume growth of Kählerian manifolds with positive curvature near infinity

In this paper, the authors proved that the order of volume growth of Kählerian manifolds with positive bisectional curvature near infinity is at least half of the real dimension (i.e., the complex dimension).