Manifolds Of Constant Curvature Research Articles

Integral formulae for codimension-one foliated Randers spaces

Integral formulae for foliated Riemannian manifolds provide obstructions for existence of foliations or compact leaves of them with given geometric properties. This paper continues our recent study and presents new integral formulae and their applications for codimension-one foliated Randers spaces. The goal is a generalization of Reeb's formula (that the total mean curvature of the leaves is zero) and its companion (that twice total second mean curvature of the leaves equals to the total Ricci curvature in the normal direction). We also extend results by Brito, Langevin and Rosenberg (that total mean curvatures of arbitrary order for a codimension-one foliated Riemannian manifold of constant curvature don't depend on a foliation). All of that is done by a comparison of extrinsic and intrinsic curvatures of the two Riemannian structures which arise in a natural way from a given Randers structure.

Extended superalgebras from twistor and Killing spinors

The basic first-order differential operators of spin geometry that are Dirac operator and twistor operator are considered. Special types of spinors defined from these operators such as twistor spinors and Killing spinors are discussed. Symmetry operators of massless and massive Dirac equations are introduced and relevant symmetry operators of twistor spinors and Killing spinors are constructed from Killing–Yano (KY) and conformal Killing–Yano (CKY) forms in constant curvature and Einstein manifolds. The squaring map of spinors gives KY and CKY forms for Killing and twistor spinors, respectively. They constitute a graded Lie algebra structure in some special cases. By using the graded Lie algebra structure of KY and CKY forms, extended Killing and conformal superalgebras are constructed in constant curvature and Einstein manifolds.

Prolongation of symmetric Killing tensors and commuting symmetries of the Laplace operator

We determine the space of commuting symmetries of the Laplace operator on pseudo-Riemannian manifolds of constant curvature and derive its algebra structure. Our construction is based on Riemannian tractor calculus, allowing us to construct a prolongation of the differential system for symmetric Killing tensors. We also discuss some aspects of its relation to projective differential geometry.

On Kakeya-Nikodym type maximal inequalities

We show that for any dimension d ≥ 3 d\ge 3 , one can obtain Wolff’s L ( d + 2 ) / 2 L^{(d+2)/2} bound on Kakeya-Nikodym maximal function in R d \mathbb R^d for d ≥ 3 d\ge 3 without the induction on scales argument. The key ingredient is to reduce to a 2-dimensional L 2 L^2 estimate with an auxiliary maximal function. We also prove that the same L ( d + 2 ) / 2 L^{(d+2)/2} bound holds for Nikodym maximal function for any manifold ( M d , g ) (M^d,g) with constant curvature, which generalizes Sogge’s results for d = 3 d=3 to any d ≥ 3 d\ge 3 . As in the 3-dimensional case, we can handle manifolds of constant curvature due to the fact that, in this case, two intersecting geodesics uniquely determine a 2-dimensional totally geodesic submanifold, which allows the use of the auxiliary maximal function.

Rigidity Theorems for Complete Sasakian Manifolds with Constant Pseudo-Hermitian Scalar Curvature

The orthogonal decomposition of the Webster curvature provides us a way to characterize some canonical metrics on a pseudo-Hermitian manifold. We derive some subelliptic differential inequalities from the Weitzenb\"ock formulas for the traceless pseudo-Hermitian Ricci tensor and the Chern-Moser tensor of Sasakian manifolds with constant pseudo-Hermitian scalar curvature and Sasakian pseudo-Einstein manifolds respectively. By means of either subelliptic estimates or maximum principle, some rigidity theorems are established to characterize Sasakian pseudo-Einstein manifolds among Sasakian manifolds with constant pseudo-Hermitian scalar curvature and Sasakian space forms among Sasakian pseudo-Einstein manifolds respectively.

Affine analogues of the Sasaki-Shchepetilov connection

AbstractFor two-dimensional manifold M with locally symmetric connection ∇ and with ∇-parallel volume element vol one can construct a flat connection on the vector bundle TM ⊕ E, where E is a trivial bundle. The metrizable case, when M is a Riemannian manifold of constant curvature, together with its higher dimension generalizations, was studied by A.V. Shchepetilov [J. Phys. A: 36 (2003), 3893-3898]. This paper deals with the case of non-metrizable locally symmetric connection. Two flat connections on TM ⊕ (ℝ × M) and two on TM ⊕ (ℝ2 × M) are constructed. It is shown that two of those connections – one from each pair – may be identified with the standard flat connection in ℝN, after suitable local affine embedding of (M,∇) into ℝN.

Stable solutions of the Yamabe equation on non-compact manifolds

We consider the Yamabe equation on a complete non-compact Riemannian manifold and study the condition of stability of solutions. If $(M^m ,g)$ is a closed manifold of constant positive scalar curvature, which we normalize to be $m(m-1)$, we consider the Riemannian product with the $n$-dimensional Euclidean space: $(M^m \times \mathbb{R}^n , g+ g_E )$. And we study, as in [2], the solution of the Yamabe equation which depends only on the Euclidean factor. We show that there exists a constant $\lambda (m,n)$ such that this solution is stable if and only if $\lambda_1 \geq \lambda (m,n)$, where $\lambda_1$ is the first positive eigenvalue of $-\Delta_g$. We compute $\lambda (m,n)$ numerically for small values of $m,n$ showing in these cases that the Euclidean minimizer is stable in the case $M=S^m$ with the metric of constant curvature. This implies that the same is true for any closed manifold with a Yamabe metric.

Generalized Wintgen inequality for statistical submanifolds in statistical manifolds of constant curvature

The Wintgen inequality (1979) is a sharp geometric inequality for surfaces in the 4-dimensional Euclidean space involving the Gauss curvature (intrinsic invariant) and the normal curvature and squared mean curvature (extrinsic invariants), respectively. De Smet et al. (Arch. Math. (Brno) 35:115–128, 1999) conjectured a generalized Wintgen inequality for submanifolds of arbitrary dimension and codimension in Riemannian space forms. This conjecture was proved by Lu (J. Funct. Anal. 261:1284–1308, 2011) and by Ge and Tang (Pac. J. Math. 237:87–95, 2008), independently. In the present paper we establish a generalized Wintgen inequality for statistical submanifolds in statistical manifolds of constant curvature.

The continuous transition of Hamiltonian vector fields through manifolds of constant curvature

We ask whether Hamiltonian vector fields defined on spaces of constant Gaussian curvature κ (spheres, for κ > 0, and hyperbolic spheres, for κ < 0) pass continuously through the value κ = 0 if the potential functions Uκ, κ ∈ ℝ, which define them satisfy the property limκ→0Uκ=U0, where U0 corresponds to the Euclidean case. We prove that the answer to this question is positive, both in the 2- and 3-dimensional cases, which are of physical interest, and then apply our conclusions to the gravitational N-body problem.

On a Classification of 4-d Gradient Ricci Solitons with Harmonic Weyl Curvature

We study a characterization of 4-dimensional (not necessarily complete) gradient Ricci solitons $(M, g, f)$ which have harmonic Weyl curvature, i.e. $\delta W=0$. Roughly speaking, we prove that the soliton metric $g$ is locally isometric to one of the following four types: an Einstein metric, the product $ \mathbb{R}^2 \times N_{\lambda}$ of the Euclidean metric and a 2-d Riemannian manifold of constant curvature ${\lambda} \neq 0$, a certain singular metric and a locally conformally flat metric. The method here is motivated by Cao-Chen's works \cite{CC1, CC2} and Derdzi\'{n}ski's study on Codazzi tensors \cite{De}. Combined with the previous results on locally conformally flat solitons, our characterization yields a new classification of 4-d complete steady solitons with $\delta W=0$. For shrinking case, it reproves the rigidity result \cite{FG, MS} in 4-d. It also helps to understand the expanding case; we now understand all 4-d non-conformally-flat ones with $\delta W=0$. We also characterize {\it locally} 4-d (not necessarily complete) gradient Ricci solitons with harmonic curvature.

Lie algebra of conformal Killing–Yano forms

We provide a generalization of the Lie algebra of conformal Killing vector fields to conformal Killing–Yano forms. A new Lie bracket for conformal Killing–Yano forms that corresponds to slightly modified Schouten–Nijenhuis bracket of differential forms is proposed. We show that conformal Killing–Yano forms satisfy a graded Lie algebra in constant curvature manifolds. It is also proven that normal conformal Killing–Yano forms in Einstein manifolds also satisfy a graded Lie algebra. The constructed graded Lie algebras reduce to the graded Lie algebra of Killing–Yano forms and the Lie algebras of conformal Killing and Killing vector fields in special cases.

On first and second eigenvalues of Riesz transforms in spherical and hyperbolic geometries

In this note we prove an analogue of the Rayleigh–Faber–Krahn inequality, that is, that the geodesic ball is a maximiser of the first eigenvalue of some convolution type integral operators, on the sphere \(\mathbb {S}^{n}\) and on the real hyperbolic space \(\mathbb {H}^{n}\). It completes the study of such question for complete, connected, simply connected Riemannian manifolds of constant sectional curvature. We also discuss an extremum problem for the second eigenvalue on \(\mathbb {H}^{n}\) and prove the Hong–Krahn–Szego type inequality. The main examples of the considered convolution type operators are the Riesz transforms with respect to the geodesic distance of the space.

Generalized Sasakian Space Forms and Riemannian Manifolds of Quasi Constant Sectional Curvature

In this paper, we show that a generalized Sasakian space form of dimension greater than three is either of constant sectional curvature; or a canal hypersurface in Euclidean or Minkowski spaces; or locally a certain type of twisted product of a real line and a flat almost Hermitian manifold; or locally a wapred product of a real line and a generalized complex space form; or an $\alpha$-Sasakian space form; or it is of five dimension and admits an $\alpha$-Sasakian Einstein structure. In particular, a local classification for generalized Sasakian space forms of dimension greater than five is obtained. A local classification of Riemannian manifolds of quasi constant sectional curvature of dimension greater than three is also given in this paper.

Osserman Lightlike Hypersurfaces on a Foliated Class of Lorentzian Manifolds

This paper deals with a family of Osserman lightlike hypersurfaces $(M_u)$ of a class of Lorentzian manifolds $\bar{M}$ such that its each null normal vector is defined on some open subset of $\bar{M}$ around $M_u$. We prove that a totally umbilical family of lightlike hypersurfaces of a connected Lorentzian pointwise Osserman manifold of constant curvature is locally Einstein and pointwise ${\cal F}-$Osserman, where our foliation approach provides the required algebraic symmetries of the induced curvature tensor. Also we prove two new characterization theorems for the family of Osserman lightlike hypersurfaces, supported by a physical example of Osserman lightlike hypersurfaces of the Schwarzschild spacetime.

On the sub-Riemannian geometry of contact Anosov flows

We investigate certain natural connections between sub-Riemannian geometry and hyperbolic dynamical systems. In particular, we study dynamically defined horizontal distributions which split into two integrable ones and ask: how is the energy of a sub-Riemannian geodesic shared between its projections onto the integrable summands? We show that if the horizontal distribution is the sum of the strong stable and strong unstable distributions of a special type of a contact Anosov flow in three dimensions, then for any short enough sub-Riemannian geodesic connecting points on the same orbit of the Anosov flow, the energy of the geodesic is shared equally between its projections onto the stable and unstable bundles. The proof relies on a connection between the geodesic equations and the harmonic oscillator equation, and its explicit solution by the Jacobi elliptic functions. Using a different idea, we prove an analogous result in higher dimensions for the geodesic flow of a closed Riemannian manifold of constant negative curvature.

Dimensions of projections of sets on Riemannian surfaces of constant curvature

We apply the theory of Peres and Schlag to obtain generic lower bounds for Hausdorff dimension of images of sets by orthogonal projections on simply connected two-dimensional Riemannian manifolds of constant curvature. As a conclusion we obtain appropriate versions of Marstrand’s theorem, Kaufman’s theorem, and Falconer’s theorem in the above geometrical settings.

On a class of complete Finsler manifolds

Here, it is shown that if a forward geodesically complete Finsler manifold admits a circle preserving change of metric then its indicatrix is conformally diffeomorphic to the Euclidean sphere Sn-1. Moreover, if the Finsler manifold is absolutely homogeneous and of scalar flag curvature then it is a Riemannian manifold of constant sectional curvature. These results provide a geometric interpretation for existence of solutions to the certain ODE on the Riemannian tangent space.

ON THE LIE DERIVATIVE OF REAL HYPERSURFACES IN ℂP2AND ℂH2WITH RESPECT TO THE GENERALIZED TANAKA-WEBSTER CONNECTION

Abstract. In this paper the notion of Lie derivative of a tensor fieldT of type (1,1) of real hypersurfaces in complex space forms with re-spect to the generalized Tanaka-Webster connection is introduced andis called generalized Tanaka-Webster Lie derivative. Furthermore, threedimensional real hypersurfaces in non-flat complex space forms whosegeneralized Tanaka-Webster Lie derivative of 1) shape operator, 2) struc-ture Jacobi operator coincides with the covariant derivative of them withrespect to any vector field X orthogonal to ξ are studied. 1. IntroductionA complex space form is an n-dimensional Kahler manifold of constant holo-morphic sectional curvature c. A complete and simply connected complex spaceform is analytically isometric to a complex projective space CP n if c > 0, or toa complex Euclidean space C n if c = 0, or to a complex hyperbolic space CH n if c < 0. The complex projective and complex hyperbolic spaces are callednon-flat complex space forms, since c 6= 0 and the symbol M

New black holes of vacuum Einstein equations with hyperscaling violation and Nil geometry horizons

In this paper, we present a new solution of the vacuum Einstein equations in five dimensions which is a static black hole with hyperscaling violation and with a three-dimensional horizon modeled by one of the eight Thurston geometries, namely the Nil geometry. This homogeneous geometry is nontrivial in the sense that it is neither of constant curvature nor a product of constant curvature manifolds. Using the Hamiltonian formalism, we identify the mass and entropy of the black hole solution. Curiously enough, in spite of the fact that the entropy turns out to be negative, the mass is positive and the first law of thermodynamics holds. We also discuss the extension in higher dimensions.

Doubly elliptic strings on the (anti) de Sitter manifold

We present a new class of elliptic-like strings on two-dimensional manifolds of constant curvature. Our solutions are related to a class of identities between Jacobi theta functions and to the geometry of the light-cone in one dimension more. We show in particular that two well-known fundamental identities among theta functions have a natural interpretation as expressing the Virasoro constraints of dS or AdS strings.