Ion Mihai

• Professor
• Professor at University of Bucharest

We establish an improved Chen inequality involving scalar curvature and mean curvature and geometric inequalities for Casorati curvatures, on slant submanifolds in a Lorentzian–Sasakian space form endowed with a semi-symmetric non-metric connection. Also, we present examples of slant submanifolds in a Lorentzian–Sasakian space form.

The geometry of submanifolds in Kähler manifolds is an important research topic. In the present paper, we study submanifolds in complex space forms admitting a semi-symmetric non-metric connection. We prove the Chen–Ricci inequality, Chen basic inequality, and a generalized Euler inequality for such submanifolds. These inequalities provide estimati...

We study Lorentzian contact and Lorentzian–Sasakian structures in Hom-Lie algebras. We find that the three-dimensional sl(2,R) and Heisenberg Lie algebras provide examples of such structures, respectively. Curvature tensor properties in Lorentzian–Sasakian Hom-Lie algebras are investigated. If v is a contact 1-form, conditions under which the Ricci...

This paper deals with statistical submanifolds and a family of statistical connections on them. The geometric structures such as the second fundamental form, curvatures tensor, mean curvature, statistical Ricci curvature and the relations among them on a statistical submanifold of a statistical manifold equipped with F-statistical connections are e...

Published as Special Issue of the Romanian Journal of Mathematics and Computer Science, vol. 13(2) (2023), available online at https://rjm-cs.utcb.ro/all-issues/

In the present article, we study submanifolds tangent to the Reeb vector field in trans-Sasakian manifolds. We prove Chen’s first inequality and the Chen–Ricci inequality, respectively, for such submanifolds in trans-Sasakian manifolds which admit a semi-symmetric non-metric connection. Moreover, a generalized Euler inequality for special contact s...

In this note we propose a new sectional curvature on a Riemannian manifold endowed with a semi-symmetric non-metric connection. A Chen-Ricci inequality is proven. Some possible applications in other fields are mentioned.

B.-Y. Chen and O. J. Garay studied pointwise slant submanifolds of almost Hermitian manifolds. By using the notion of pointwise slant submanifolds, we investigate the geometry of pointwise semi-slant submanifolds and their warped products in Sasakian and cosymplectic manifolds. We prove that there exist no proper pointwise semi-slant warped product...

in the present article, we consider bi-slant submanifolds in trans-Sasakian generalized Sasakianspace forms. Specifically, we establish both the Chen first inequality and the Chen-Ricci inequality on suchsubmanifolds. We provide an example of bi-slant submanifold.Keywords: Chen first invariant, squared mean curvature, Ricci curvature, trans-Sasakia...

A Norden manifold is a complex manifold endowed with a pair of Norden metrics. We consider dual connections on such manifolds and study their geometric properties. One example of a complex statistical connection is provided. Mathematics Subject Classification (2010): 53C05, 53C15.

In the present article, we consider submanifolds in golden Riemannian manifolds with constant golden sectional curvature. On such submanifolds, we prove geometric inequalities for the Casorati curvatures. The submanifolds meeting the equality cases are also described.

In this paper, we address the study of the Kobayashi–Nomizu type and the Yano type connections on the tangent bundle TM equipped with the Sasaki metric. Then, we determine the curvature tensors of these connections. Moreover, we find conditions under which these connections are torsion-free, Codazzi, and statistical structures, respectively, with r...

The purpose of this paper is to find some conditions under which the tangent bundle TM has a dualistic structure. Then, we introduce infinitesimal affine transformations on statistical manifolds and investigate these structures on a special statistical distribution and the tangent bundle of a statistical manifold too. Moreover, we also study the mu...

A class of variable‐order fractional optimal control problems (VO‐FOCPs) is solved by applying Müntz‐Legendre wavelets. Different from classical wavelets (such as Legendre and Chebyshev), the Müntz‐Legendre wavelets have an extra parameter representing the fractional order; therefore, they provide more reliable results for certain fractional calcul...

In this paper, we improve the Chen first inequality for special contact slant submanifolds and Legendrian submanifolds, respectively, in (α,β) trans-Sasakian generalized Sasakian space forms endowed with a semi-symmetric metric connection.

Chen’s first inequality for statistical submanifolds in Hessian manifolds of constant Hessian curvature was obtained by B.-Y. Chen et al. Other particular cases of Chen inequalities in a statistical setting were given by different authors. The objective of the present article is to establish the general Chen inequalities for statistical submanifold...

In this paper, we obtain a geometric inequality for warped product pointwise semi-slant submanifolds of complex space forms endowed with a semi-symmetric metric connection and discuss the equality case of this inequality. We provide some applications concerning the minimality and compactness of such submanifolds.

We consider a statistical connection ∇ on an almost complex manifold with (pseudo-) Riemannian metric, in particular the Norden metric. We investigate almost Norden (statistical) manifolds under the condition that the almost complex structure J is ∇-recurrent. We provide one example of a complex statistical connection.

The objective of the present article is to prove two geometric inequalities for submanifolds in S-space forms. First, we establish inequalities for the generalized normalized δ-Casorati curvatures for bi-slant submanifolds in S-space forms and then we derive the generalized Wintgen inequality for Legendrian and bi-slant submanifolds in the same amb...

In 1990, B.-Y. Chen defined the slant submanifolds in complex manifolds as a natural generalization of holomorphic and totally real submanifolds [9]. Examples of slant submanifolds in \(\mathbb {C}^{2}\) and \(\mathbb {C}^{4}\) were given by Chen and Tazawa [20, 21]. This study was extended by Lotta [28] in almost contact geometry and further inves...

A bi-warped product of the form: $M=N_T \times_{f_1}N^{n_{1}}_\perp\times_{f_2} N^{n_{2}}_\theta$ in a contact metric manifold is called a CRS bi-warped product, where $N_T,\, N^{n_{1}}_\perp$ and $N^{n_{2}}_θ$ are invariant, anti-invariant and proper pointwise slant submanifolds, respectively. First, we prove that there are no proper CRS bi-warpe...

An open problem in reliability theory is that of finding all the coefficients of the reliability polynomial associated with particular networks. Because reliability polynomials can be expressed in Bernstein form (hence linked to binomial coefficients), it is clear that an extension of the classical discrete Pascal’s triangle (comprising all the bin...

In Differential Geometry, Kähler and Sasaki manifolds and their submanifolds are probably the most studied geometric objects, because of their interesting properties. In particular, the behavior of submanifolds in complex space forms and Sasakian space forms was investigated by many geometers.

In the present paper, we investigate some properties of the distributions involved in the definition of a CR-statistical submanifold. The characterization of a CR-product in holomorphic statistical manifolds is given. By using an optimization technique, we establish a relationship between the Ricci curvature and the squared norm of the mean curvatu...

Introducing left invariant almost para-contact metric structures on Hom-Lie groups (Hom-Lie algebras), we study the normal and para-Sasakian conditions on them. The left-invariant curvature tensor properties of left-invariant para-Sasakian Hom-Lie groups are investigated. Some examples are commented as support of the obtained results.

Curvature invariants are the most natural invariants, with a wide application in science and engineering. A known condition for a Riemannian manifold to admit a minimal immersion in any Euclidean space is Ric≤0. In order to find other obstructions, one needs to introduce new types of Riemannian invariants, different in nature from classical ones (C...

We give a simple proof of the Chen inequality involving the Chen invariant δ(k) of submanifolds in Riemannian space forms. We derive Chen’s first inequality and the Chen–Ricci inequality. Additionally, we establish a corresponding inequality for statistical submanifolds.

We give a simple proof of the Chen inequality for the Chen invariant d(2, . . . , 2) (k terms) of submanifolds in Riemannian space forms

In the present paper, weinvestigate some properties of the distributions involved in the definition of a CR-statistical submanifold. The characterization of a CR-product in holomorphic statistical manifolds is given. By using an optimization technique, we establish a relationship between the Ricci curvature and the squared norm of the mean curvatur...

In the geometry of submanifolds, Chen inequalities represent one of the most important tool to find relationships between intrinsic and extrinsic invariants; the aim is to find sharp such inequalities. In this paper we establish an optimal inequality for the Chen invariant δ ( 2 , 2 ) on Lagrangian submanifolds in quaternionic space forms, regarded...

We establish Chen inequality for the invariant δ(2, 2) on statistical submanifolds in Hessian manifolds of constant Hessian curvature. Recently, in cooperation with Chen, we proved a Chen first inequality for such submanifolds. The present authors previously initiated the investigation of statistical submanifolds in Hessian manifolds of constant He...

In 2008, Chen and Dillen obtained a sharp estimation for the squared norm of the second fundamental form of multiply warped CR-submanifold \(M=M_1\times _{f_2}M_2\times \ldots \times _{f_k}M_k\) in an arbitrary Kähler manifold \({\tilde{M}}\) such that \(M_1\) is a holomorphic submanifold and \(M_\perp =_{f_2}M_2\times \cdots \times _{f_k}M_k\) is...

In the present article we initiate the study of submanifolds in normal complex contact metric manifolds. We define invariant and anti-invariant ( C C -totally real) submanifolds in such manifolds and start the study of their basic properties. Also, we establish the Chen first inequality and Chen inequality for the invariant δ ( 2 , 2 ) for C C -tot...

In this paper, we consider Mθ, a pointwise slant submanifold and prove that every bi-warped product M⊥×f1MT×f2Mθ in a locally product Riemannian manifold satisfies a general inequality: ‖σ‖2≥n2‖∇→T(lnf1)‖2+n3cos2θ‖∇→θ(lnf2)‖2,where n2=dim(MT),n3=dim(Mθ) and σ is the second fundamental form and ∇T(lnf1) and ∇θ(lnf2) are the gradient components along...

The study of statistical submanifolds in Hessian manifolds of constant Hessian curvature was started by two of the present authors. We continue this work and establish a Chen first inequality for such submanifolds.

Chen (2001) initiated the study of CR-warped product submanifolds in Kaehler manifolds and established a general inequality between an intrinsic invariant (the warping function) and an extrinsic invariant (second fundamental form). In this paper, we establish a relationship for the squared norm of the second fundamental form (an extrinsic invariant...

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. In the present paper we obtain a Wintgen inequality for statistical surfaces.

In this paper, we study bi-warped product submanifolds of the form: $M=N_T \times_{f_1}N^{n_{1}}_\perp\times_{f_2} N^{n_{2}}_\theta$ in a contact metric manifold $\widetilde M$, where $N_T,\, N^{n_{1}}_\perp$ and $N^{n_{2}}_\theta$ are invariant, anti-invariant and proper pointwise slant submanifolds of $\widetilde M$, respectively. We simply calle...

The theory of \(\delta \)-invariants was initiated by Chen (Arch Math 60:568–578, 1993) in order to find new necessary conditions for a Riemannian manifold to admit a minimal isometric immersion in a Euclidean space. Chen (Int J Math 23(3):1250045, 2012) defined a CR \(\delta \)-invariant \(\delta (D)\) for anti-holomorphic submanifolds in complex...

We consider statistical submanifolds of Hessian manifolds of constant Hessian curvature. For such submanifolds we establish a Euler inequality and a Chen-Ricci inequality with respect to a sectional curvature of the ambient Hessian manifold.

In this paper, we introduce the notion of warped product skew CR-submanifolds in Kenmotsu manifolds. We obtain several results on such submanifolds. A characterization for skew CR-submanifolds is obtained. Furthermore, we establish an inequality for the squared norm of the second fundamental form of a warped product skew CR-submanifold M 1 × f M ⊥...

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 inequalit...

The isotropic space is a special ambient space obtained from the Euclidean space by substituting the usual Euclidean distance with the isotropic distance. In the present paper, we establish a method to calculate the second fundamental form of surfaces in the isotropic 4-space. Further, we classify some surfaces (spherical product surfaces and Amino...

Recently, B.-Y. Chen and O. J. Garay studied pointwise slant submanifolds of almost Hermitian manifolds. By using the notion of pointwise slant submanifolds, we investigate the geometry of pointwise semi-slant submanifolds and their warped products in Sasakian and cosymplectic manifolds. We prove that there exist no proper pointwise semi-slant warp...

Legendrian submanifolds in Sasakian space forms play an important role in contact geometry. Defever et al. (Boll Unione Mat Ital B 7(11):365–374, 1997) established the first Chen inequality for C-totally real submanifolds in Sasakian space forms. In this article, we improve this first Chen inequality for Legendrian submanifolds in Sasakian space fo...

We survey recent results on CR-submanifolds in complex space forms and contact CR-submanifolds in Sasakian space forms, including a few contributions of the present authors. The Ricci curvature and k-Ricci curvature of such submanifolds are estimated in terms of the squared mean curvature. A Wintgen-type inequality for totally real surfaces in comp...

The normal scalar curvature conjecture, also known as the DDVV conjecture, was formulated by De Smet, Dillen, Verstraelen and Vrancken in 1999. It was proven recently by Lu [19] and by Ge and Tang [15] independently. Recently we obtained DDVV inequalities, also known as generalized Wintgen inequalities, for Lagrangian submanifolds in complex space...

In this paper, we study the behaviour of submanifolds in statistical manifolds of constant curvature. We investigate curvature properties of such submanifolds. Some inequalities for submanifolds with any codimension and hypersurfaces of statistical manifolds of constant curvature are also established.

In this paper, we study the behaviour of submanifolds in statistical manifolds of constant curvature. We investigate curvature properties of such submanifolds. Some inequalities for submanifolds with any codimension and hypersurfaces of statistical manifolds of constant curvature are also established.

The normal scalar curvature conjecture, also known as the DDVV conjecture, was formulated by De Smet et al. (1999). It was proven recently by Lu (2011) and by Ge and Tang (2008) independently. We obtain the DDVV inequality, also known as the generalized Wintgen inequality, for Lagrangian submanifolds in complex space forms. Some applications are gi...

The generalized Wintgen inequality was conjectured by De Smet, Dillen, Verstraelen and Vrancken in 1999 for submanifolds in real space forms. It is also known as the DDVV conjecture. It was proven recently by Lu (2011) and by Ge and Tang (2008), independently. The present author established a generalized Wintgen inequality for Lagrangian sub-manifo...

We interrelate the concept of a torse forming vector field and the concepts of exterior concurrent and quasi-exterior concurrent vector fields. Different second order properties of a torse forming vector field T are studied, as for instance it is proved that any torse forming is a quasi-exterior concurrent vector field. We obtain a necessary and su...

The geometry of submanifolds, in particular in Hermitian manifolds, is an important topic of research in Differential Geometry.

An isometric immersion f: M ® [(M)\tilde]\phi : M \rightarrow \tilde{M} of a manifold M into a Kähler manifold is called purely real if the complex structure J on [(M)\tilde]\tilde{M} carries the tangent bundle of M into a transversal bundle. In an earlier article, the first author discovered a general optimal inequality for purely real surfaces in...

An isometric immersion φ: M → ˜M of a manifold M into a K¨ahler manifold is called purely real if the complex structure J on ˜M carries the tangent bundle of M into a transversal bundle. In an earlier article, the first author discovered a general optimal inequality for purely real surfaces in a complex space form ˜M 2(4ε). He also studied purely r...

From J-action point of views, slant surfaces are the simplest and the most natural surfaces of a (Lorentzian) Kähler surface ($$\tilde M,\tilde g $$, J). Slant surfaces arise naturally and play some important roles in the studies of surfaces of Kähler surfaces (see, for instance, [13]). In this article, we classify quasi-minimal slant surfaces in t...

B.Y. Chen and the present author have defined some contact Riemannian invariants for almost contact metric manifolds analogous to Chen invariants. Sharp inequalities between these contact Riemannian invariants and the squared mean curvature for almost contact metric manifolds in a real space form were established. The equality case of these inequal...

The notion of a slant surface in a Lorentzian Kähler manifolds was introduced in [2]. Following R. Rosca (see, for instance, [3]), a surface M in a Lorentzian manifold is called quasi-minimal if its mean curvature vector is light-like at each point. B.Y. Chen and F. Dillen [1] classified quasi-minimal Lagrangian surfaces in Lorentzian complex space...

The notion of a slant surface in a Lorentzian K ̈ahler manifolds was introduced in [2]. Following R. Rosca (see, for instance, [3]), a surface M in a Lorentzian manifold is called quasi-minimal if its mean curvature vector is light-like at each point. B.Y. Chen and F. Dillen [1] classified quasi-minimal Lagrangian surfaces in Lorentzian complex spa...

Chen and Ogiue completely classified totally umbilical submanifolds in a non-flat complex-space-form. However, the classification problem of pseudo-umbilical submanifolds in a non-flat complex-space-form is still open. Very recently, Chen introduced the notion of Lagrangian H-umbilical submanifolds which is the simplest totally real submanifolds ne...

Chen and Ogiue completely classified totally umbilical submanifolds in a non-flat complex-space-form. However, the classification problem of pseudo-umbilical submanifolds in a non-flat complex-space-form is still open. Very recently, Chen introduced the notion of Lagrangian H-umbilical submanifolds which is the simplest totally real submanifolds ne...

The geometry of complex manifolds, in particular Kaehler manifolds, is an important research topic in Differential Geometry. The chapter presents the basic notions and certain important results in complex differential geometry. It defines complex and almost complex manifolds and gives standard examples. Locally, a Kaehlerian metric differs from the...

Recently, Chen established a general sharp inequality for warped products in real space forms. As applications, he obtained obstructions to minimal isometric immersions of warped products into real space forms. Afterwards, Matsumoto and one of the present authors proved the Sasakian version of this inequality. In the present paper, we obtain sharp...

We deal with a locally conformal cosymplectic manifold M(φ,Ω,ξ,η,g) admitting a conformal contact quasi-torse-forming vector field T. The presymplectic 2-form Ω is a locally conformal cosymplectic 2-form. It is shown that T is a 3-exterior concurrent vector field. Infinitesimal transformations of the Lie algebra of ∧M are investigated. The Gauss ma...

Recently, Chen studied warped products which are CR-submanifolds in Kaehler manifolds and established general sharp inequalities for CR-warped products in Kaehler manifolds. In the present paper, we obtain sharp estimates for the squared norm of the second fundamental form (an extrinsic invariant) in terms of the warping function for contact CR-war...

First, we define some contact Riemannian invariants for almost contact metric manifolds analogues to those invariants introduced in [5, 6]. We then establish sharp inequalities between these contact Riemannian invariants and the squared mean curvature for almost contact Riemannian manifolds in a real space form. We also investigate almost contact R...

Roughly speaking, an ideal immersion of a Riemannian manifold into a space form is an isometric im-mersion which produces the least possible amount of tension from the ambient space at each point of the submanifold. Recently, Chen studied Lagrangian submanifolds in complex space forms which are ideal. He proved that such submanifolds are minimal. H...

Roughly speaking, an ideal immersion of a Riemannian manifold into a space form is an isometric immersion which produces the least possible amount of tension from the ambient space at each point of the submanifold. Recently, B.-Y. Chen studied Lagrangian submanifolds in complex space forms which are ideal. He proved that such submanifolds are minim...

Roughly speaking, an ideal immersion of a Riemannian man-ifold into a space form is an isometric immersion which produces the least possible amount of tension from the ambient space at each point of the sub-manifold. Recently, B.-Y. Chen studied Lagrangian submanifolds in complex space forms which are ideal. He proved that such submanifolds are min...

Recently, B.-Y. Chen studied warped products which are CR-submanifolds in Kaehler manifolds and established general sharp inequalities for CR-warped products in Kaehler manifolds. Afterwards, I. Hasegawa and the present author obtained a sharp inequality for the squared norm of the second fundamental form (an extrinsic invariant) in terms of the wa...

We deal with a Riemannian manifoldM carrying a pair of skew symmetric conformal vector fields (X, Y). The existence of such a pairing is determined by an exterior differential system in involution (in the sense of Cartan). In this case,M is foliated by 3-dimensional totally geodesic submanifolds. Additional geometric properties are proved.

One of the present authors established a sharp relationship between the Ricci curvature and the squared mean curvature for a submanifold in a Sasakian space form. In this article, we derive a sharp estimate of the maximum Ricci curvature function for a special contact slant submanifold M in a Sasakian space form M (c), in terms of the main extrinsi...

Recently, B.-Y. Chen studied warped products which are CR-submanifolds in Kaehler manifolds and established general sharp inequalities for CR-warped products in Kaehler manifolds. In the present paper, we obtain a sharp inequality for the squared norm of the second fundamental form (an extrinsic invariant) in terms of the warping function for conta...

Recently, B.-Y. Chen obtained an inequality for slant surfaces in complex space forms. Further, B.-Y. Chen and one of the present authors proved the non-minimality of proper slant surfaces in non-flat complex space forms. In the present paper, we investigate 3-dimensional proper contact slant submanifolds in Sasakian space forms. A sharp inequality...

B.-Y. Chen obtained an inequality for slant surfaces in complex space forms. Further, B.-Y. Chen and one of the present authors proved the non-minimality of proper slant surfaces in non-flat complex space forms. In the present paper, we investigate 3-dimensional proper contact slant submanifolds in Sasakian space forms. A sharp inequality is obtain...

Roughly speaking, an ideal immersion of a Riemannian manifold into a space form is an isometric immersion which produces the least possible amount of tension from the ambient space at each point of the submanifold. Recently, B.-Y. Chen [Dillen, Franki (ed.) et al., Geometry and topology of submanifolds, VI. Proceedings of the international meetings...

We prove that for a given Riemaannian metric g on an 2n-di-mensional differentiable manifold M̃ which admits an n-dimensional foliation F, there exists an almost Hermitian structure (double-struck G sign, double-struck F sign) on M̃ determined by the pairing (g, F). In particular, we investigate the case when it is almost Kählerian or Kählerian.

B.-Y. Chen established a sharp relationship between the Ricci curvature and the squared mean curvature for a submanifold in a Riemannian space form with arbi-trary codimension. The Lagrangian version of this inequality was proved by the same author. In this article, we obtain a sharp estimate of the Ricci tensor of a slant submanifold M in a comple...

We obtain certain inequalities involving several intrinsic invariants namely scalar curvature, Ricci curvature and $k$-Ricci curvature, and main extrinsic invariant namely squared mean curvature for submanifolds in a locally conformal almost cosymplectic manifold with pointwise constant $% \phi $-sectional curvature. Applying these inequalities we...

We deal with a Riemannian manifold M carrying a pair of skew symmetric Killing vector fields (X, Y). The existence of such a pairing is determined by an exterior differential system in involution (in the sense of Cartan). In this case, M is foliated by 3-dimensional submanifolds of constant scalar curvature. Additional geometric properties are prov...

We deal with a normal conformal cosymplectic (2m+ 1)-dimensional manifold M(Ω, ∅, η. ξ, g), with Ω as the fundamental 2-form and η the Reeb covector field. Following [1], there exists a unique vector field X λ such that $$\Omega ({X_\lambda }) = i{x_\lambda }\Omega = d\lambda - \xi (\lambda )\eta $$ Xλ is called a Hamiltonian vector field. It is pr...

B. Y. Chen [Kodai Math. J. 4, 399–417 (1981; Zbl 0481.53046)] established a sharp relationship between the Ricci curvature and the squared mean curvature for a submanifold in a Riemannian space form with arbitrary codimension. Recently Ximin Liu [Arch. Math. (Brno), 38, 297–305 (2002; Zbl 1090.53052)] obtained results on Ricci curvature of a totall...

Recently, Chen established a general sharp inequality for warped products in real space forms. As applications, he obtained obstructions to minimal isometric immersions of warped products into real space forms. In the present paper, we obtain sharp inequalities for warped products isomet- rically immersed in Sasakian space forms. Some applications...

Recently, Chen established a sharp relationship between the Ricci curvature and the squared mean curvature for a submanifold in a Riemannian space form with arbitrary codimension. Afterwards, we dealt with similar problems for submanifolds in complex space forms. In the present paper, we obtain sharp inequalities between the Ricci curvature and the...

In 1999, Chen established a sharp relationship between the Ricci curvature and the squared mean curvature for a submanifold in a Riemannian space form with arbitrary codimension. Similar problems for submanifolds in complex space forms were studied by Matsumoto et al. In this paper, we obtain sharp relationships between the Ricci curvature and the...

We deal with a 2m-dimensional Riemannian manifold (M,g) structured by an affine connection and a vector field 𝒯, defining a 𝒯-parallel connection. It is proved that 𝒯 is both a torse forming vector field and an exterior concurrent vector field. Properties of the curvature 2-forms are established. It is shown that M is endowed with a conformal symp...