Riemannian Geometry of Contact and Symplectic Manifolds
Chapters (3)
Let P and M be C
∞ manifolds, π : P → M a C
∞ map of P onto M and G a Lie group acting on P to the right. Then (P, G, M) is called a principal G-bundle if
1.
G acts freely on P,
2.
π(p
1) = π(p
2) if and only if there exists g ∈ G such that p
1g = p
2,
3.
P is locally trivial over M, i.e., for every m ∈ M there exists a neighborhood U of m and a map F
u
: π-1(U) → G such that F
u
(pg) = (F
u
(p))g and such that the map Ψ : π-1(U) → U × G taking p to (π(p), F
u
(p)) is a diffeomorphism.
By a contact manifold we mean a C
∞ manifold M
2n+1 together with a 1-form η such that η ∧ (dη)n
≠ 0. In particular η ∧ (dη)n
≠ 0 is a volume element on M so that a contact manifold is orientable. Also dη has rank 2n on the Grassmann algebra ∧ T
m
*
M at each point m ∈ M and thus we have a 1-dimensional subspace, {X ∈ T
m
M|dη(X, T
m
M) = 0}, on which η ≠ 0 and which is complementary to the subspace on which η = 0. Therefore choosing ξ
m
in this subspace normalized by η(ξ
m
) = 1 we have a global vector field ξ satisfying $$ d\eta \left( {\xi ,X} \right) = 0,\;\eta \left( \xi \right) = 1 $$.
While the study of complex contact manifolds is almost as old as the modern theory of real contact manifolds, the subject has received much less attention and as many examples are now appearing in the literature, especially twistor spaces over quaternionic Kähler manifolds (e.g., LeBrun [1991], [1995], Moroianu and Semmelmann [1994], Salamon [1982], Ye [1994]), the time is ripe for another look at the subject. As an indication of this interest we note, for example, the following result of Moroianu and Semmelmann [1994] that on a compact spin Kähler manifold M of positive scalar curvature and complex dimension 4l + 3, the following are equivalent: (i) M is a Kähler—Einstein manifold admitting a complex contact structure, (ii) M is the twistor space of a quaternionic Kähler manifold of positive scalar curvature, (iii) M admits Kählerian Killing spinors. LeBrun [1995] proves that a complex contact manifold of positive first Chern class, i.e., a Fano contact manifold, is a twistor space if and only if it admits a Kähler-Einstein metric and conjectures that every Fano contact manifold is a twistor space.
... which implies that η(X) = g(ξ, X). Moreover, an almost contact metric manifold M is said to be a Sasakian manifold if and only if [4], Theorem 6.3 ...
... , y n , z) denotes the Cartesian coordinates on R 2n+1 . Then (R 2n+1 , φ, η, ξ, g) is a Sasakian manifold (see [4] for more details). ...
... A plane section in T M is called a φ-section if there exists a vector X ∈ T M orthogonal to ξ such that {X, φX} span the section. The sectional curvature K(X, φ), denoted H(X), is called φ-sectional curvature (see [4] for more details and references therein). Then we have (5.14) ...
... According to Blair [2], a smooth Riemannian manifold (M, g) of dimension (2n+ 1) is classified as an almost contact metric manifold if it admits a global 1-form η, called a contact form, such that η ∧(dη) 2 = 0, a (1, 1) tensor field φ, a characteristic vector field ξ, and an indefinite metric g.These must satisfy the following relations: ...
... for all vector fields ω 1 and ω 2 on M . When an almost contact manifold M 2n+1 (φ, ξ, η, g) is equipped with a compatible metric g, it is referred to as an almost contact metric manifold (as defined by Blair [2]). Let us consider an almost contact metric manifold M 2n+1 (φ, ξ, η, g) where the following conditions hold true: ...
... Here, [φ, φ] represents the Nijenhuis tensor associated with φ (for more details see [2]) Definition 2.1. On an almost contact metric manifold M , a vector field ω 1 is said to be contact vector field if there exist a smooth function f such that L ω1 ξ = f ξ. ...
In this paper, we aim to investigate the properties of an almost -Ricci-Bourguignon soliton (almost R-B-S for short) on a Kenmotsu manifold (K-M). We start by proving that if a Kenmotsu manifold (K-M) obeys an almost R-B-S, then the manifold is -Einstein. Furthermore, we establish that if a -nullity distribution, where , has an almost -Ricci-Bourguignon soliton (almost R-B-S), then the manifold is Ricci flat. Moreover, we establish that if a K-M has almost -Ricci-Bourguignon soliton gradient and the vector field preserves the scalar curvature r, then the manifold is an Einstein manifold with a constant scalar curvature given by . Finaly, we have given en example of a almost R-B-S gradient on the Kenmotsu manifold.
... Let M be a (2n + 1)-dimensional manifold endowed with an almost contact structure (ϕ, ξ, η), i.e., ϕ is a tensor field of type (1, 1), ξ is a vector field, and η is a 1-form satisfying [1] (3.1) ...
... The concepts of nearly cosymplectic and nearly Sasakian manifolds were defined in [1] for Riemannian metric. We adapt the same definitions in the case of semi-Riemannian settings. ...
... where H T X and H N X are the tangential and normal components of HX, respectively. Moreover, M is indefinite α-Sasakian if and only if H vanishes identically on M (see [1]). As an example, we have the following. ...
We introduce invariant rigged null hypersurfaces of indefinite almost contact manifolds, by paying attention to those of indefinite nearly α-Sasakian manifolds. We prove that, under some conditions, there exist leaves of the integrable screen distribution of the ambient manifolds admitting nearly α-Sasakian structures.
... The Reeb vector field of a contact manifold plays a fundamental role in the study of the Riemannian geometry of a contact metric manifold. More detailed information about the contact metric manifolds and the Reeb vector fields can be found in [3,14]. Besides, consider several types of almost contact metric manifolds. ...
We study the case when a unit vector field ξ on a Riemannian manifold (M,g) defines an isometric embedding ξ:(M,g)→(T1M, G), where G is the Riemannian g-natural metric. The main goal is to find conditions under which the submanifold ξ(M)⊂(T1M, G) can be totally geodesic. It is proved that the Reeb vector field of a K-contact metric structure on M gives rise to totally geodesic ξ(M) if and only if the structure is Sasakian. As a by-product, we find the expression for the second fundamental form of ξ(M)⊂(T1M, G).
... g(φX, φY ) = g(X, Y ) + η(X)η(Y ), (2) η(X) = εg(X, ξ), ...
In this paper we study a semi-Riemannian submersion from Lorentzian (para)almost contact manifolds and find necessary and sufficient conditions for the characteristic vector field to be vertical or horizontal. We also obtain decomposition theorems for an anti-invariant semi-Riemannian submersion from Lorentzian (para)Sasakian manifolds onto a Lorentzian manifold.
... Note that p plays the role of the spectral parameter, since u p ≡ 0 by assumption. These nonisospectral Lax pairs have a connection to contact geometry: the vector field X h formally looks exactly like a contact vector field corresponding to a contact Hamiltonian h on a contact 3-manifold with local coordinates x, z, p and the associated contact one-form dz + pdx (see [29,30] and the references therein). ...
This paper introduces a (3 + 1)-dimensional dispersionless integrable system, utilizing a Lax pair involving contact vector fields, in alignment with methodologies presented by Sergyeyev in 2014. Significantly, it is shown that the proposed system serves as an integrable (3 + 1)-dimensional generalization of the well-studied (2 + 1)-dimensional dispersionless Davey–Stewartson system. This way, an interesting new example on integrability in higher dimensions is presented, with potential applications in analyzing three-dimensional nonlinear waves across various fields, including oceanography, fluid dynamics, plasma physics, and nonlinear optics. Importantly, the integrable nature of the system suggests that established techniques like the study of symmetries, conservation laws, and Hamiltonian structures could be applicable.
... where ω(X) = g(ψ, X) for all X vector field on M . For more background on almost contact metric manifolds, we refer to [7,5,10,14,17,21]. ...
The present paper is devoted to a class of manifolds which admit an f-structure with 2-dimensional parallelizable kernel. Such manifolds are called 4-dimensional almost bi-contact metric manifolds; they carry a locally conformal almost Kähler structure. We give some classifications and prove their fundamental properties, then we deduce some properties about the complex manifolds associated with them. We show the existence of such manifolds by giving some non-trivial examples. Finally, we establish an interesting class and construct a concrete example.
... For more background on almost contact metric manifolds, we recommend the references [1,4,9]. ...
The present paper is devoted to three-dimensional C_12-manifolds (defined by D. Chinea and C. Gonzalez), which are never normal. We study their fundamental properties and give concrete examples. As an application, we study such structures on three-dimensional Lie groups.
... An odd-dimensional Riemannian manifold (M, g) is called an almost contact metric manifold [138] if there exist a (1, 1)-tensor field φ, a vector field ξ and a one-form η, such that ...
One of the most fundamental interests in submanifold theory is to establish simple relationships between the main extrinsic invariants and the main intrinsic invariants of submanifolds and find their applications. In this respect, the first author established, in 1993, a basic inequality involving the first δ-invariant, δ(2), and the squared mean curvature of submanifolds in real space forms, known today as the first Chen inequality or Chen's first inequality. Since then, there have been many papers dealing with this inequality.
The purpose of this article is, thus, to present a comprehensive survey on recent developments on this inequality performed by many geometers during the last three decades.
In this paper, we introduce a geometric description of contact Lagrangian and Hamiltonian systems on Lie algebroids in the framework of contact geometry, using the theory of prolongations. We discuss the relation between Lagrangian and Hamiltonian settings through a convenient notion of Legendre transformation. We also discuss the Hamilton-Jacobi problem in this framework and introduce the notion of a Legendrian Lie subalgebroid of a contact Lie algebroid.
Our main aim is to introduce Clairaut slant submersions in complex geometry. We give the notion of Clairaut slant submersions from almost Hermitian manifolds onto Riemannian manifold in this article. We obtain some basic results on discussed submersions. Furthermore, we provide some examples to explore the geometry of Clairaut slant submersions.
In this paper, we consider generalized -Ricci soliton on compact and non-compact K-contact manifolds. First, it is established that if a compact K-contact metric g represents a generalized -Ricci soliton with the potential vector field V as a Jacobi field along the Reeb vector field , then it is -Einstein. Next, we characterize a generalized -Ricci soliton when the potential vector field is either a contact vector field, or a projective vector field.
This paper focuses on some geometrical and physical properties of a conformal η-Ricci soliton (Cη-RS) on a four-dimension Lorentzian Para-Sasakian (LP-S) manifold. The first section presents an introduction to Cη-RS on LP-S manifolds, followed by a discussion of preliminary ideas about the LP-Sasakian manifold. In the subsequent sections, we establish several results pertaining to four-dimension LP-S manifolds that exhibit Cη-RS. Additionally, we consider certain conditions associated with Cη-RS on four-dimension LP-S manifolds. Besides these geometrical points of view, we consider this soliton in a perfect fluid spacetime and obtain some interesting physical properties. Finally, we present a case study of a Cη-RS on a four-dimension LP-S manifold.
All invariant contact metric structures on tangent sphere bundles of each compact rank-one symmetric space are obtained explicitly, distinguishing for the orthogonal case those that are K-contact, Sasakian or 3-Sasakian. Only the tangent sphere bundle of complex projective spaces admits 3-Sasakian metrics and there exists a unique orthogonal Sasakian-Einstein metric on Furthermore, there is a unique invariant contact metric that is Einstein, in fact Sasakian-Einstein, on tangent sphere bundles of spheres and real projective spaces. Each invariant contact metric, Sasakian, Sasakian-Einstein or 3-Sasakian structure on the unit tangent sphere of any compact rank-one symmetric space is extended, respectively, to an invariant almost Kähler, Kähler, Kähler Ricci-flat or hyperKähler structure on the punctured tangent bundle.
In this study, we consider bi-f-harmonic Legendre curves on -trans-Sasakian generalized Sasakian space form. We provide the necessary and sufficient conditions for a Legendre curve to be bi-f-harmonic on -trans-Sasakian generalized Sasakian space form without any restrictions by a main theorem. Afterward, we investigate these conditions under nine different cases. As a result of these investigations, we obtain the original theorems and corollaries as well as the nonexistence theorems. We perform these investigations according to the and functions from the curvature tensor of the -trans-Sasakian generalized Sasakian space form, the curvature and torsion of the bi-f-harmonic Legendre curve, and finally, the positions of the basis vectors relative to each other.
We give explicit parametrizations for all the homogeneous Riemannian structures on model spaces of Thurston geometry. As an application, we give all the homogeneous contact metric structures on 3-dimensional Sasakian space forms.
We study self-gravitating solutions of three-dimensional massive gravity coupled to the Yang–Mills–Chern–Simons gauge theory. Among these, there is a family of asymptotically Warped-Anti de Sitter black holes that come to generalize previous solutions found in the literature and studied in the context of WAdS 3 /CFT 2 . We also present self-gravitating solutions to three-dimensional Einstein–Yang–Mills theory, as well other self-gravitating solutions in the presence of higher-curvature terms.
In this paper, we construct almost contact metric structures on a three-dimensional Riemannian manifold equipped with an almost Ricci soliton. Then, we give the techniques necessary to define the nature of such structures. Concrete examples are given.
We study on metallic shaped contact hypersurfaces of Kaehler manifolds. We show that a metallic shaped (?,?)-contact metric hypersurface of a Kaehler manifold has constant mean curvature. As a special case, we also consider product shaped Sasakian hypersurfaces of Kaehler manifolds.
In this paper, we provide new and simpler proofs of two theorems of Gluck and Harrison on contact structures induced by great circle or line fibrations. Furthermore, we prove that a geodesic vector field whose Jacobi tensor is parallel along flow lines (e.g. if the underlying manifold is locally symmetric) induces a contact structure if the ‘mixed’ sectional curvatures are nonnegative, and if a certain nondegeneracy condition holds. Additionally, we prove that in dimension three, contact structures admitting a Reeb flow which is either periodic, isometric, or free and proper, must be universally tight. In particular, we generalise an earlier result of Geiges and the author, by showing that every contact form on R3 whose Reeb vector field spans a line fibration is necessarily tight. Furthermore, we provide a characterisation of isometric Reeb vector fields. As an application, we recover a result of Kegel and Lange on Seifert fibrations spanned by Reeb vector fields, and we classify closed contact 3-manifolds with isometric Reeb flows (also known as R-contact manifolds) up to diffeomorphism.
We prove that if contact strongly pseudo-convex integrable CR-manifold admits a *-Ricci soliton where the soliton vector Z is contact, then the Reeb vector field ? is an eigenvector of the Ricci operator at each point if and only if ? is constant. Then we study contact strongly pseudo-convex integrable CR-manifold such that 1 is a almost *-Ricci soliton with potential vector field Z collinear with ?. To this end, we prove that if a 3-dimensional contact metric manifoldMwith Q? = ?Q which admits a gradient almost *-Ricci soliton, then either M is flat or f is constant.
In this paper, we thoroughly study [Formula: see text]-Ricci–Bourguignon almost soliton and gradient [Formula: see text]-Ricci–Bourguignon almost soliton in the paracontact geometry, precisely, on [Formula: see text]-paracontact and para-Sasakian manifolds. Here, we prove that if the metric [Formula: see text] of the [Formula: see text]-paracontact manifold endows a [Formula: see text]-Ricci–Bourguignon almost soliton with the nonzero potential vector field [Formula: see text] parallel to [Formula: see text], then the manifold is an Einstein with Einstein constant [Formula: see text]. Next, we show that if a para-Sasakian manifold represents a gradient [Formula: see text]-Ricci–Bourguignon almost soliton, then the manifold is an Einstein with constant scalar curvature [Formula: see text]. We also discuss [Formula: see text]-Ricci–Bourguignon almost soliton on [Formula: see text]-paracontact manifold.
In this paper, we characterize static perfect fluid spacetimes on almost co-Kähler manifolds. At first, it is shown that if a [Formula: see text]-almost co-Kähler manifold [Formula: see text] is the spatial factor of a static perfect fluid spacetime, then [Formula: see text] is either a [Formula: see text]-almost co-Kähler manifold or Ricci flat. Next, we consider the case for a 3-dimensional co-Kähler manifold [Formula: see text] as the spatial factor of a static perfect fluid spacetime.
In this paper quasi contact metric manifolds, which are conformally flat are studied. We prove that a conformally flat quasi contact metric manifold M2n+1 with n≥3 whose characteristic vector field is Killing, has constant curvature +1+1 and thus is Sasakian. Also we consider conformally flat quasi contact metric manifolds on which the Ricci operator Q commutes with ϕ.
The main purpose of the paper is to study an almost gradient Ricci–Bourguignon soliton (RB soliton) within the framework of [Formula: see text]-contact manifolds and [Formula: see text]-contact manifolds. First, we prove that if complete [Formula: see text]-contact manifold endows a gradient RB soliton, then the manifold is compact Sasakian and isometric to unit sphere [Formula: see text]. Next, we show that if a complete contact metric satisfies an almost RB soliton with a non-zero potential vector field is collinear with the Reeb vector field [Formula: see text] and the Reeb vector field [Formula: see text] acting as an eigenvector of the Ricci operator, then it is compact Einstein Sasakian and the potential vector field is a constant multiple of the Reeb vector field [Formula: see text]. Lastly, we prove that if the metric of a non-Sasakian [Formula: see text]-contact manifold is an almost gradient RB soliton, then it is flat in dimension 3 and in higher dimensions it is locally isometric to [Formula: see text].
I. Mihai obtained an inequality relating intrinsic normalised scalar curvature and extrinsic squared mean curvature and normalised normal curvature of Legendrian submanifolds Mn in Sasakian space forms eM2n+1(c). In this paper, for the class of generalised Wintgen ideal Legendrian submanifolds Mn of Sasakian space form eM2n+1(c), we study relationship between some properties concerning their Deszcz symmetry and their Roter type.
We introduce screen generic lightlike submanifolds of indefinite cosymplectic manifolds. We investigate the integrability of various distributions and prove a characterization theorem of such lightlike submanifolds in a cosymplectic space form. We study contact totally umbilical screen generic lightlike submanifolds and minimal screen generic lightlike submanifolds. We also give examples.
The goal of this paper is two-fold: firstly, we give a necessary and sufficient condition for the existence of a transverse Einstein metric on a given co-oriented contact manifold . Secondly, we apply this result to obtain important consequences. In particular, we give another proof of a theorem by Boyer and Galicki (Sasakian Geometry. Oxford Mathematical Monographs. Oxford University Press, Oxford (2008). xii+613pp.) on solutions to Goldberg’s conjecture for -Einstein K-contact manifolds. An example illustrating our construction result is provided.
This paper deals with the study of generalized Cauchy–Riemann screen pseudo-slant lightlike submanifolds of indefinite Kaehler manifolds giving a characterization theorem with some nontrivial examples of such submanifolds. Integrability conditions of distributions [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] on GCR screen pseudo-slant lightlike submanifolds of indefinite Kaehler manifolds have been obtained. Furthermore, we obtain necessary and sufficient conditions for foliations determined by the above distributions to be totally geodesic.
In this paper, we define a new kind of curve called N-slant curve whose principal normal vector field makes a constant angle with the Reeb vector field ξ in Sasakian 3-manifolds. Then, we give some characterizations of N-slant curves in Sasakian 3-manifolds and we obtain some properties of the curves in R3(−3) . Moreover, we investigate the conditions of C−parallel and C−proper mean curvature vector fields along N-slant curves in Sasakian 3-manifolds. Finally, we study N-slant curves of type AW(k).
The present paper aims to study the complete lifts of quarter-symmetric non-metric -connection from a 3-dimensional non-cosymplectic quasi-Sasakian manifold to its tangent bundle and establish specific curvature properties of such connection on the tangent bundle.
In this paper, we study some triviality and rigidity results of Riemann soliton. First, we derive some sufficient conditions for which an almost Riemann soliton is trivial. In particular, we prove that any compact almost Riemann soliton with constant scalar curvature has constant sectional curvature. Next, we prove some rigidity results for gradient Riemann solitons. Precisely, we prove that a non-trivial gradient Riemann soliton is locally isometric to a warped product (I×F,dt2+f(t)2gF), where ∇σ≠0.
In this paper, we discuss the reduction of symplectic Hamiltonian systems by scaling and standard symmetries which commute. We prove that such a reduction process produces a so‐called Kirillov Hamiltonian system. Moreover, we show that if we reduce first by the scaling symmetries and then by the standard ones or in the opposite order, we obtain equivalent Kirillov Hamiltonian systems. In the particular case when the configuration space of the symplectic Hamiltonian system is a Lie group , which coincides with the symmetry group, the reduced structure is an interesting Kirillov version of the Lie–Poisson structure on the dual space of the Lie algebra of . We also discuss a reconstruction process for symplectic Hamiltonian systems which admit a scaling symmetry. All the previous results are illustrated in detail with some interesting examples.
We construct a Wick-type deformation quantization of contact metric manifolds. The construction is fully canonical and involves no arbitrary choice. Unlike the case of symplectic or Poisson manifolds, not every classical observable on a general contact metric manifold can be promoted to a quantum one due to possible obstructions to quantization. We prove, however, that all these obstructions disappear for Sasakian manifolds.
Recently, there are a great deal of work done which connects the Legendrian isotopic problem with contact invariants. The isotopic problem of Legendre curve in a contact 3-manifold was studied via the Legendrian curve shortening flow which was introduced and studied by K. Smoczyk. On the other hand, in the SYZ Conjecture, one can model a special Lagrangian singularity locally as the special Lagrangian cones in . This can be characterized by its link which is a minimal Legendrian surface in the 5-sphere. Then in these points of view, we will focus on the existence of the long-time solution and asymptotic convergnce along the Legendrian mean curvature flow in the (2n+1)-dimensional -Einstein Sasakian manifolds under the suitable stability condition due to the Thomas-Yau conjecture.
The almost contact metric structure that we have on a real hypersurface M in the complex quadric Q m = SO m +2 / SO m SO 2 allows us to define, for any nonnull real number k , the k -th generalized Tanaka-Webster connection on M , ∇ ^ ( k ) . Associated to this connection, we have Cho and torsion operators F X ( k ) and T X ( k ) , respectively, for any vector field X tangent to M . From them and for any symmetric operator B on M , we can consider two tensor fields of type (1,2) on M that we denote by B F ( k ) and B T ( k ) , respectively. We classify real hypersurfaces M in Q m for which any of those tensors identically vanishes, in the particular case of B being the structure Lie operator L ξ on M .
The object of the present paper is to study the notion of three dimensional locally nearly recurrent generalized (k, µ)−space forms and nearly quasi-concircular ϕ−recurrent generalized (k, µ)− space form M(f 1 , f 2 , f 3 , f 4 , f 5 , f 6).
. The object of the present paper is to study Quasi-concircularly flat and φ−quasi-concircularly flat generalized Sasakian-space-forms. Also, we consider generalized Sasakian-space-forms satisfying the condition P(ξ, X).Ve = 0, Ve(ξ, X).P = 0, and Ve(ξ, X).Ve = 0 and we obtain some important results. Finally, we give an example.
In this paper, we give some characterizations by considering ∗-Ricci soliton as a Kenmotsu metric. We prove that if a Kenmotsu manifold represents an almost ∗-Ricci soliton and the potential vector field [Formula: see text] is a Jacobi along the Reeb vector field, then it is a steady ∗-Ricci soliton. Next, we show that a Kenmotsu matric endowed an almost ∗-Ricci soliton which is Einstein metric if it is [Formula: see text]-Einstein or the potential vector field [Formula: see text] is collinear to the Reeb vector field or [Formula: see text] is an infinitesimal contact transformation.
A bstract
A generalization of the 4d Chern-Simons theory action introduced by Costello and Yamazaki is presented. We apply general arguments from symplectic geometry concerning the Hamiltonian action of a symmetry group on the space of gauge connections defined on a 4d manifold and construct an action functional that is quadratic in the moment map associated to the group action. The generalization relies on the use of contact 1-forms defined on non-trivial circle bundles over Riemann surfaces and mimics closely the approach used by Beasley and Witten to reformulate conventional 3d Chern-Simons theories on Seifert manifolds. We also show that the path integral of the generalized theory associated to integrable field theories of the PCM type, takes the canonical form of a symplectic integral over a subspace of the space of gauge connections, turning it a potential candidate for using the method of non-Abelian localization. Alternatively, this new quadratic completion of the 4d Chern-Simons theory can also be deduced in an intuitive way from manipulations similar to those used in T-duality. Further details on how to recover the original 4d Chern-Simons theory data, from the point of view of the Hamiltonian formalism applied to the generalized theory, are included as well.
The main objective of this paper is to study semi-symmetric almost -cosymplectic three-manifolds. We present basic formulas for almost -cosymplectic manifolds. Using curvature properties, we obtain some necessary and sufficient conditions on semi-symmetric almost -cosymplectic three-manifolds. We obtain the main results under an additional condition. The paper concludes with two illustrative examples.
Zhang (2021), Luo and Yin (2022) initiated the study of Lagrangian submanifolds satisfying ∇*T = 0 or ∇*∇*T = 0 in ℂn or ℂℙn, where and is the Lagrangian trace-free second fundamental form. They proved several rigidity theorems for Lagrangian surfaces satisfying ∇*T = 0 or ∇*∇*T = 0 in ℂ2 under proper small energy assumption and gave new characterization of the Whitney spheres in ℂ2. In this paper, the authors extend these results to Lagrangian submanifolds in ℂn of dimension n ≥ 3 and to Lagrangian submanifolds in ℂℙn.
We construct and study the intrinsic sub-Laplacian, defined outside the set of characteristic points, for a smooth hypersurface embedded in a contact sub-Riemannian manifold. We prove that, away from characteristic points, the intrinsic sub-Laplacian arises as the limit of Laplace–Beltrami operators built by means of Riemannian approximations to the sub-Riemannian structure using the Reeb vector field. We carefully analyse three families of model cases for this setting obtained by considering canonical hypersurfaces embedded in model spaces for contact sub-Riemannian manifolds. In these model cases, we show that the intrinsic sub-Laplacian is stochastically complete and in particular, that the stochastic process induced by the intrinsic sub-Laplacian almost surely does not hit characteristic points.
We consider gradient generalized -Ricci solitons on contact metric manifolds and prove some rigidity results under certain conditions that appear naturally on Sasakian structure. First, we prove that a gradient generalized -Ricci soliton is rigid if it is radially flat and has Codazzi type Ricci tensor. Next, it is proved that a compact contact metric manifold of dimension admitting a non-trivial gradient generalized -Ricci soliton with vanishing radial Weyl tensor and satisfying is locally isometric to a unit sphere .
We consider real hypersurfaces M in complex projective space equipped with both the Levi–Civita and generalized Tanaka–Webster connections. For any nonnull constant k and any symmetric tensor field of type (1, 1) L on M , we can define two tensor fields of type (1, 2) on M , L F ( k ) and L T ( k ) , related to both connections. We study the behaviour of the structure operator ϕ with respect to such tensor fields in the particular case of L=A L = A , the shape operator of M , and obtain some new characterizations of ruled real hypersurfaces in complex projective space.
The object of this article is to study a new class of almost contact metric structures which are integrable but non normal. Secondly, we explain a method of construction for normal manifold starting from a non-normal but integrable manifold. Illustrative examples are given.
In this paper, we obtain some classification results of three-dimensional non-coK?hler almost coK?hler manifold M whose Reeb vector field is strongly normal unit vector field with ?(???h?) = 0, for which the *-Ricci tensor is of Codazzi-type or M satisfies the curvature condition Q* ? R = 0.
Let M be a real hypersurface of a complex projective space. For any operator B on M and any nonnull real number k , we can define two tensor fields of type (1,2) on M , B F ( k ) and B T ( k ) . We will classify real hypersurfaces in complex projective space for which B F ( k ) and B T ( k ) either take values in the maximal holomorphic distribution D or are parallel to the structure vector field ξ , in the particular case of B=A B = A , where A denotes the shape operator of M . We also introduce the concept of A F ( k ) and A T ( k ) being D -recurrent and classify real hypersurfaces such that either A F ( k ) or A T ( k ) are D -recurrent.
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