Showing papers in "Differential Geometry and Its Applications in 2011"
TL;DR: In this article, a metric quasi-Einstein metric is defined, where the m -Bakry-Emery Ricci tensor is a constant multiple of the metric tensor.
Abstract: We call a metric quasi-Einstein if the m -Bakry–Emery Ricci tensor is a constant multiple of the metric tensor. This is a generalization of Einstein metrics, it contains gradient Ricci solitons and is also closely related to the construction of the warped product Einstein metrics. We study properties of quasi-Einstein metrics and prove several rigidity results. We also give a splitting theorem for some Kahler quasi-Einstein metrics.
209 citations
TL;DR: In this article, a new class of Finsler metrics called general (α, β ) -norms are introduced, which are defined by a Riemannian metric and a 1-form.
Abstract: In this paper, the geometric meaning of ( α , β ) -norms is made clear. On this basis, a new class of Finsler metrics called general ( α , β ) -metrics are introduced, which are defined by a Riemannian metric and a 1-form. These metrics not only generalize ( α , β ) -metrics naturally, but also include some metrics structured by R. Bryant. The spray coefficients formula of some kinds of general ( α , β ) -metrics is given and the projective flatness is also discussed.
94 citations
TL;DR: In this paper, the authors define the thin fundamental Gray 3-groupoid S3(M) of a smooth manifold M and define (by using differential geometric data) 3-dimensional holonomies, to be smooth strict Gray 3groupoid maps S3M! C(H), where H is a 2-crossed module of Lie groups.
Abstract: We define the thin fundamental Gray 3-groupoid S3(M) of a smooth manifold M and define (by using differential geometric data) 3-dimensional holonomies, to be smooth strict Gray 3-groupoid maps S3(M) ! C(H), where H is a 2-crossed module of Lie groups and C(H) is the Gray 3groupoid naturally constructed from H. As an application, we define Wilson 3-sphere observables.
66 citations
TL;DR: In this paper, the existence of a calibrated G2-structure on a Lie algebra of dimension seven is investigated and a classification of the nilpotent Lie algebras that admit such structures is given.
Abstract: We introduce obstructions to the existence of a calibrated G2-structure on a Lie algebra g of dimension seven, not necessarily nilpotent. In particular, we prove that if there is a Lie algebra epimorphism from g to a six-dimensional Lie algebra h with kernel contained in the center of g , then h has a symplectic form. As a consequence, we obtain a classification of the nilpotent Lie algebras that admit a calibrated G2-structure.
46 citations
TL;DR: In this paper, Chen-Ricci inequality and improved Chen Ricci inequality for curvature-like tensors were presented and applied to Lagrangian and Kaehlerian slant submanifolds of complex space forms.
Abstract: We present Chen–Ricci inequality and improved Chen–Ricci inequality for curvature like tensors. Applying our improved Chen–Ricci inequality we study Lagrangian and Kaehlerian slant submanifolds of complex space forms, and C -totally real submanifolds of Sasakian space forms.
41 citations
TL;DR: The necessary and sufficient condition on a Randers space for the existence of a measure for which Shen's S-curvature vanishes everywhere is given in this paper. And if it exists, such a measure coincides with the Busemann-Hausdorff measure up to a constant multiplication.
Abstract: We give a necessary and sufficient condition on a Randers space for the existence of a measure for which Shen’s S-curvature vanishes everywhere. Moreover, if it exists, such a measure coincides with the Busemann-Hausdorff measure up to a constant multiplication.
31 citations
TL;DR: In this article, the authors generalize the Hermitian curvature flow introduced and studied in Streets and Tian (2011) to the almost complex case and show that it can be used for almost complex curvature flows.
Abstract: In the present paper we generalize the Hermitian curvature flow introduced and studied in Streets and Tian (2011) [6] to the almost complex case.
30 citations
TL;DR: In this article, a weakly complex Berwald metric is defined, and the complex Wrona metric on the Euclidean sphere S 2 n − 1 ⊂ C n is obtained.
Abstract: In this paper, we give a definition of weakly complex Berwald metric and prove that, (i) a strongly convex weakly Kahler–Finsler metric F on a complex manifold M is a weakly complex Berwald metric iff F is a real Berwald metric; (ii) assume that a strongly convex weakly Kahler–Finsler metric F is a weakly complex Berwald metric, then the associated real and complex Berwald connections coincide iff a suitable contraction of the curvature components of type ( 2 , 0 ) of the complex Berwald connection vanish; (iii) the complex Wrona metric in C n is a fundamental example of weakly complex Berwald metric whose holomorphic curvature and Ricci scalar curvature vanish identically. Moreover, the real geodesic of the complex Wrona metric on the Euclidean sphere S 2 n − 1 ⊂ C n is explicitly obtained.
28 citations
TL;DR: In this article, the authors studied scalar and symmetric 2-form valued universal curvature identities and established the Gauss-Bonnet theorem using heat equation methods, and gave a new proof of a result of Kuzʼmina and Labbi concerning the Euler-Lagrange equations of the GAuss−Bonnet integral.
Abstract: We study scalar and symmetric 2-form valued universal curvature identities. We use this to establish the Gauss–Bonnet theorem using heat equation methods, to give a new proof of a result of Kuzʼmina and Labbi concerning the Euler–Lagrange equations of the Gauss–Bonnet integral, and to give a new derivation of the Euh–Park–Sekigawa identity.
28 citations
TL;DR: In this paper, it was shown that the obstruction to the existence of a global conserved current is the difference of two conceptually independent cohomology classes: one coming from using the symmetries of the Euler-Lagrange morphism and the other from the system of local Noether currents.
Abstract: We investigate globality properties of conserved currents associated with local variational problems admitting global Euler–Lagrange morphisms. We show that the obstruction to the existence of a global conserved current is the difference of two conceptually independent cohomology classes: one coming from using the symmetries of the Euler–Lagrange morphism and the other from the system of local Noether currents.
28 citations
TL;DR: In this article, the first non-zero eigenvalue p 1 of Steklov problem on M in terms of the r-th mean curvatures of its boundary ∂M was derived.
Abstract: Let M be a compact submanifold with boundary of a Euclidean space or a Sphere. In this paper, we derive an upper bound for the first non-zero eigenvalue p 1 of Steklov problem on M in terms of the r-th mean curvatures of its boundary ∂M. The upper bound obtained is sharp.
TL;DR: In this article, Cheng et al. studied and gave a characterization of locally dually flat (α,β)-metrics on an n-dimensional manifold M (n⩾3), which generalizes some results in Cheng et.
Abstract: The notion of locally dually flat Finsler metrics are originated from information geometry. Some special locally dually flat Finsler metrics had been studied in Cheng et al. (2009) (in press) [6] and Xia (in press) [10] respectively. As we know, (α,β)-metrics defined by a Riemannian metric α and a 1-form β form an important class of Finsler metrics. In this paper, we study and give a characterization of locally dually flat (α,β)-metrics on an n-dimensional manifold M (n⩾3), which generalizes some results in Cheng et al. (2009) (in press) [6] and Xia (in press) [10].
TL;DR: In this article, the authors investigate constant mean curvature complete vertical graphs in a warped product, which is supposed to satisfy an appropriated convergence condition, and obtain rigidity theorems concerning to such graphs.
Abstract: 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.
TL;DR: In this paper, it was shown that two smooth nearby Riemannian metrics can be glued interpolating their scalar curvature, and that the resulting smooth metric is the same as the starting ones outside the gluing region and has scalare curvature interpolating between the original ones.
Abstract: We show that two smooth nearby Riemannian metrics can be glued interpolating their scalar curvature. The resulting smooth metric is the same as the starting ones outside the gluing region and has scalar curvature interpolating between the original ones. One can then glue metrics while maintaining inequalities satisfied by the scalar curvature. We also glue asymptotically Euclidean metrics to Schwarzschild ones and the same for asymptotically Delaunay metrics, keeping bounds on the scalar curvature, if any. This extends the Corvino gluing near infinity to non-constant scalar curvature metrics.
TL;DR: In this paper, a comparison theorem on Ricci curvature for projectively related Finsler metrics was obtained for a class of projectively flat sprays, which particularly showed that there exist many isotropic sprays that cannot be induced by any (even singular) FINSler metrics.
Abstract: This paper studies some properties of projective changes in spray and Finsler geometry. Firstly, it obtains a comparison theorem on Ricci curvature for projectively related Finsler metrics. Secondly, it studies the properties of a class of projectively flat sprays, which particularly shows that there exist many isotropic sprays that cannot be induced by any (even singular) Finsler metrics.
TL;DR: In this article, contact structures with associated pseudo-Riemannian metrics were studied, emphasizing their relationship and analogies with respect to the Riemannians case, and the present author focused here on contact Lorentzian structures.
Abstract: Contact structures with associated pseudo-Riemannian metrics were studied by D. Perrone and the present author (2010) in [8] . We focus here on contact Lorentzian structures, emphasizing their relationship and analogies with respect to the Riemannian case.
TL;DR: The double tetrahedron is the triangulation of the three-sphere obtained by gluing together two congruent triangulations along their boundaries, and the Yamabe problem on a general piecewise flat manifold is studied in this paper.
Abstract: The double tetrahedron is the triangulation of the three-sphere gotten by gluing together two congruent tetrahedra along their boundaries. As a piecewise flat manifold, its geometry is determined by its six edge lengths, giving a notion of a metric on the double tetrahedron. We study notions of Einstein metrics, constant scalar curvature metrics, and the Yamabe problem on the double tetrahedron, with some reference to the possibilities on a general piecewise flat manifold. The main tool is analysis of Reggeʼs Einstein–Hilbert functional, a piecewise flat analogue of the Einstein–Hilbert (or total scalar curvature) functional on Riemannian manifolds. We study the Einstein–Hilbert–Regge functional on the space of metrics and on discrete conformal classes of metrics.
TL;DR: In this paper, the authors prove a criterion for an isometric action of a Lie group on a Riemannian manifold to be polar if and only if the slice representation at the fixed point is polar and the section is the tangent space of an embedded totally geodesic submanifold.
Abstract: We prove a criterion for an isometric action of a Lie group on a Riemannian manifold to be polar. From this criterion, it follows that an action with a fixed point is polar if and only if the slice representation at the fixed point is polar and the section is the tangent space of an embedded totally geodesic submanifold. We apply this to obtain a classification of polar actions with a fixed point on symmetric spaces.
TL;DR: Para-Kahler Lie algebras which decompose as the sum of two abelian Lagrangian subalgesas are studied in this article, where several constructions and an inductive description of such Lie algesbras are provided.
Abstract: Para-Kahler Lie algebras which decompose as the sum of two abelian Lagrangian subalgebras are studied. We propose several constructions and provide an inductive description of such Lie algebras. The curvatures of the para-Kahler metric are computed and sufficient conditions to ensure flatness or Ricci-flatness are given. The Lie algebras for which the para-Kahler metric is Einstein and non-Ricci-flat are completely characterized.
TL;DR: In this article, a complete classification of non-degenerate affine hypersurfaces in R 4 with parallel cubic form with respect to the Levi-Civita connection of the affine metric is given.
Abstract: We study Lorentzian affine hypersurfaces in R n + 1 with parallel cubic form with respect to the Levi-Civita connection of the affine metric. As main result, a complete classification of such non-degenerate affine hypersurfaces in R 4 is given.
TL;DR: In this paper, the classification result of real hypersurfaces with constant principal curvatures in nonflat complex space forms and whose Hopf vector field has nontrivial projection onto two eigenspaces of the shape operator was presented.
Abstract: We present the motivation and current state of the classification problem of real hypersurfaces with constant principal curvatures in complex space forms. In particular, we explain the classification result of real hypersurfaces with constant principal curvatures in nonflat complex space forms and whose Hopf vector field has nontrivial projection onto two eigenspaces of the shape operator. This constitutes the following natural step after Kimura and Berndtʼs classifications of Hopf real hypersurfaces with constant principal curvatures in complex space forms.
TL;DR: In this article, critical metrics for the squared L 2 -norm functionals of the curvature tensor, the Ricci tensor and the scalar curvature were studied by making use of a curvature identity on 4-dimensional Riemannian manifolds.
Abstract: We study critical metrics for the squared L 2 -norm functionals of the curvature tensor, the Ricci tensor and the scalar curvature by making use of a curvature identity on 4-dimensional Riemannian manifolds.
TL;DR: In this paper, the singularities of great circular surfaces in 3-sphere were studied from the view point of projective differential geometry (i.e., SO(4)invariant geometry).
Abstract: In this paper, we consider a special class of the surfaces in 3-sphere dened by oneparameter families of great circles. We give a generic classication of singularities of such surfaces and investigate the geometric meanings from the view point of spherical geometry. In this paper we investigate a special class of surfaces in 3-sphere which are called great circular surfaces. We say that a surface in 3-sphere is a great circular surface if it is given by a oneparameter family of great circles (cf., §4). On the other hand, there appeared two kinds of curvatures in the previous theory of surfaces in 3-sphere, One is called the extrinsic Gauss curvature Ke and another is the intrinsic Gauss curvature KI. The intrinsic Gauss curvature is nothing but the Gauss curvature defined by the induced Riemannian metric on the surface. The relation between these curvatures is known that Ke = KI − 1. We can show that an extrinsic flat surface is (at least locally) parametrized as a great circular surface (cf., Theorem 3.3). Such a surface is an extrinsic flat great circular surface (briefly, we call an E-flat great circular surface). This is one of the motivation to investigate great circular surfaces. In Euclidean space, surfaces with the vanishing Gauss curvature are developable surfaces which belong to a special class of ruled surfaces [5, 6]. Therefore, the notion of great circular surfaces is one of the analogous notions with ruled surfaces in 3-sphere. In this paper, we study geometric properties and singularities of great circular surfaces. However, there is the canonical double covering π : S 3 −→ RP 3 onto the projective space. A great circle corresponds to a projective line in RP 3 , so that the singularities of great circular surfaces are the same as those of ruled surfaces. There are a lot of researches on developable surfaces in R 3 ⊂ RP 3 from the view point of Projective differential geometry [2, 4, 12, 16]. We investigate the singularities of great circular surfaces from the view point of spherical geometry (i.e, SO(4)invariant geometry). For any smooth curve A : I −→ SO(4) in the rotation group SO(4), we can define a parametrization FA of a great circular surface M = Image FA in 3-sphere. We can easily show
TL;DR: A rigidity theorem for a statistical hypersurface of Hesse-Einstein type is given in as discussed by the authors, where the authors show that it is possible to obtain a HESSIAN as discussed by the authors.
Abstract: A rigidity theorem for a statistical hypersurface of Hesse–Einstein type is given.
TL;DR: In this paper, the authors classify all surfaces with constant mean curvature that either are invariant by a 1-parameter group of isometries or are the product of two planar curves.
Abstract: Among the eight geometries of Thurston, Sol3 is the space with the smallest number of isometries, for example, there are no rotations. In this work, we classify all surfaces with constant mean curvature that either are invariant by a 1-parameter group of isometries or are the product of two planar curves.
TL;DR: In this paper, complete simply connected minimal surfaces with a prescribed coordinate function were constructed, and it was shown that these surfaces are dense in the space of all minimal surfaces in general.
Abstract: In this paper we construct complete simply connected minimal surfaces with a prescribed coordinate function. Moreover, we prove that these surfaces are dense in the space of all minimal surfaces with this coordinate function (with the topology of the smooth convergence on compact sets).
TL;DR: In this paper, the authors discuss two generalizations of the inverse problem of the calculus of variations, one in which a given mechanical system can be brought into the form of Lagrangian equations with non-conservative forces of a generalized Rayleigh dissipation type, the other leading to Lagrangians with so-called gyroscopic forces.
Abstract: We discuss two generalizations of the inverse problem of the calculus of variations, one in which a given mechanical system can be brought into the form of Lagrangian equations with non-conservative forces of a generalized Rayleigh dissipation type, the other leading to Lagrangian equations with so-called gyroscopic forces. Our approach focusses primarily on obtaining coordinate-free conditions for the existence of a suitable non-singular multiplier matrix, which will lead to an equivalent representation of a given system of second-order equations as one of these Lagrangian systems with non-conservative forces.
TL;DR: In this paper, the Riemannian geometry of almost α-Kenmotsu manifolds is analyzed, focusing on local symmetries and on some vanishing conditions for the curvature.
Abstract: We analyze the Riemannian geometry of almost α-Kenmotsu manifolds, focusing on local symmetries and on some vanishing conditions for the Riemannian curvature. If the characteristic vector field of an almost α-Kenmotsu structure belongs to the so-called (κ,μ)′-nullity distribution, κ<−α2, then the Riemannian curvature is completely determined. These manifolds provide a special case of a wider class of almost α-Kenmotsu manifolds, for which an operator h′ associated to the structure is η-parallel and has constant eigenvalues. All these manifolds are locally warped products. Finally, we give a local classification of almost α-Kenmotsu manifolds, up to D-homothetic deformations. Under suitable conditions, they are locally isomorphic to Lie groups.
TL;DR: In this article, a non-Riemannian quantity H defined by the S-curvature was studied and it was shown that H is closely related to S-Curvature.
Abstract: Article history: In this paper, we study a new non-Riemannian quantity H defined by the S-curvature. We find that the non-Riemannian quantity is closely related to S-curvature. We characterize Randers metrics of almost isotropic S-curvature if and only if they have almost vanishing H-curvature. Furthermore, the Randers metrics actually have zero S-curvature if and only if they have vanishing H-curvature.
TL;DR: In this paper, a general framework for reduction of symplectic Q-manifolds via graded group actions is presented, where the homological structure on the acting group is a multiplicative multivector field.
Abstract: We present a general framework for reduction of symplectic Q-manifolds via graded group actions. In this framework, the homological structure on the acting group is a multiplicative multivector field.