In the present chapter we deal with harmonic forms and holomorphic forms and vector fields on com... more In the present chapter we deal with harmonic forms and holomorphic forms and vector fields on compact Vaisman manifolds using the method in [275]. One of the main results is a partial answer to Vaisman’s conjectures:A compact g.H. manifold has an odd first Betti number.We shall also study the relation between holomorphic and Killing vector fields and give a certain answer to the question: How many l.c.K. metrics exist on a compact g.H. manifold?
Publisher Summary This chapter discusses fundamental matters such as the Dirichlet energy and ten... more Publisher Summary This chapter discusses fundamental matters such as the Dirichlet energy and tension tensor of a unit tangent vector field on a Riemannian manifold, first and second variation formulae, and the harmonic vector fields system. The study of the weak solutions to this system (existence and local properties) is missing from the present day mathematical literature. Various instances are investigated where harmonic vector fields occur and to generalizations. Any unit vector field that is a harmonic map is also a harmonic vector field. The study of harmonic map system is more appropriate on a Hermitian manifold and that results in Hermitian harmonic maps to be useful in studying rigidity of complete Hermitian manifolds.
This book gathers contributions by respected experts on the theory of isometric immersions betwee... more This book gathers contributions by respected experts on the theory of isometric immersions between Riemannian manifolds, and focuses on the geometry of CR structures on submanifolds in Hermitian manifolds. CR structures are a bundle theoretic recast of the tangential Cauchy Riemann equations in complex analysis involving several complex variables. The book covers a wide range of topics such as Sasakian geometry, Kaehler and locally conformal Kaehler geometry, the tangential CR equations, Lorentzian geometry, holomorphic statistical manifolds, and paraquaternionic CR submanifolds. Intended as a tribute to Professor Aurel Bejancu, who discovered the notion of a CR submanifold of a Hermitian manifold in 1978, the book provides an up-to-date overview of several topics in the geometry of CR submanifolds. Presenting detailed information on the most recent advances in the area, it represents a useful resource for mathematicians and physicists alike
The first example in the preceding section led to the study of locally conformal Kahler manifolds... more The first example in the preceding section led to the study of locally conformal Kahler manifolds with a parallel Lee form. These are calledVaisman manifolds. I. Vaisman, to whom the notion is due (cf. [269] and [2751] adopts the terminologygeneralized Hopf (g.H.) manifolds.However, the term g.H. manifold is sometimes used to label a different generalization of complex Hopf manifolds (i.e. E. Brieskorn & A. Van de Ven’s g.H. manifolds H a n (b),[45], cf. our Section 3.7). In the following, we use both terminologies interchangeably. The example of an Inoue surface with the Tricerri metric shows that g.H. manifolds form a proper subset of the set of all l.c.K. manifolds.
A fundamental problem in l.c.K. geometry is to decide which l.c.K. manifolds admit some globally ... more A fundamental problem in l.c.K. geometry is to decide which l.c.K. manifolds admit some globally defined Kahler metric. We may stateTheorem 2.1(Cf. [273])Let (M2n,J,g) be a compact l.c.K. manifold. Then (M2nJ,g)is g.c.K. if and only if there is some global Kahler metric on M2n.
In the present chapter we deal with harmonic forms and holomorphic forms and vector fields on com... more In the present chapter we deal with harmonic forms and holomorphic forms and vector fields on compact Vaisman manifolds using the method in [275]. One of the main results is a partial answer to Vaisman’s conjectures:A compact g.H. manifold has an odd first Betti number.We shall also study the relation between holomorphic and Killing vector fields and give a certain answer to the question: How many l.c.K. metrics exist on a compact g.H. manifold?
Publisher Summary This chapter discusses fundamental matters such as the Dirichlet energy and ten... more Publisher Summary This chapter discusses fundamental matters such as the Dirichlet energy and tension tensor of a unit tangent vector field on a Riemannian manifold, first and second variation formulae, and the harmonic vector fields system. The study of the weak solutions to this system (existence and local properties) is missing from the present day mathematical literature. Various instances are investigated where harmonic vector fields occur and to generalizations. Any unit vector field that is a harmonic map is also a harmonic vector field. The study of harmonic map system is more appropriate on a Hermitian manifold and that results in Hermitian harmonic maps to be useful in studying rigidity of complete Hermitian manifolds.
This book gathers contributions by respected experts on the theory of isometric immersions betwee... more This book gathers contributions by respected experts on the theory of isometric immersions between Riemannian manifolds, and focuses on the geometry of CR structures on submanifolds in Hermitian manifolds. CR structures are a bundle theoretic recast of the tangential Cauchy Riemann equations in complex analysis involving several complex variables. The book covers a wide range of topics such as Sasakian geometry, Kaehler and locally conformal Kaehler geometry, the tangential CR equations, Lorentzian geometry, holomorphic statistical manifolds, and paraquaternionic CR submanifolds. Intended as a tribute to Professor Aurel Bejancu, who discovered the notion of a CR submanifold of a Hermitian manifold in 1978, the book provides an up-to-date overview of several topics in the geometry of CR submanifolds. Presenting detailed information on the most recent advances in the area, it represents a useful resource for mathematicians and physicists alike
The first example in the preceding section led to the study of locally conformal Kahler manifolds... more The first example in the preceding section led to the study of locally conformal Kahler manifolds with a parallel Lee form. These are calledVaisman manifolds. I. Vaisman, to whom the notion is due (cf. [269] and [2751] adopts the terminologygeneralized Hopf (g.H.) manifolds.However, the term g.H. manifold is sometimes used to label a different generalization of complex Hopf manifolds (i.e. E. Brieskorn & A. Van de Ven’s g.H. manifolds H a n (b),[45], cf. our Section 3.7). In the following, we use both terminologies interchangeably. The example of an Inoue surface with the Tricerri metric shows that g.H. manifolds form a proper subset of the set of all l.c.K. manifolds.
A fundamental problem in l.c.K. geometry is to decide which l.c.K. manifolds admit some globally ... more A fundamental problem in l.c.K. geometry is to decide which l.c.K. manifolds admit some globally defined Kahler metric. We may stateTheorem 2.1(Cf. [273])Let (M2n,J,g) be a compact l.c.K. manifold. Then (M2nJ,g)is g.c.K. if and only if there is some global Kahler metric on M2n.
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