A new class of almost complex structures

The purpose of this paper is to introduce a new class of almost complex structures J on a Riemannian manifold M by using a certain identity for the relationship between the tensor F _i ^j of J and the Riemann curvature tensor R _hijk of M. This class contains the Kählerian structures, and its relationship with some known classes of almost Hermitian structures defined by similar identities is discussed. For convenience we call each structure of this new class an almost C-structure, and a manifold with an almost C-structure an almost C-manifold. We obtain an analogue of F. Schur's theorem concerning the holomorphic sectional curvature of an almost Hermitian C-manifold, and some sufficient conditions for an almost Hermitian C-manifold to be Kählerian. We show that these results are also true for a manifold with a complex structure.     

Locally symmetric immersions

We use reflections with respect to submanifolds and related geometric results to develop, inspired by the work of Ferus and other authors, in a unified way a local theory of extrinsic symmetric immersions and submanifolds in a general analytic Riemannian manifold and in locally symmetric spaces. In particular we treat the case of real and complex space forms and study additional relations with holomorphic and symplectic reflections when the ambient space is almost Hermitian. The global case is also taken into consideration and several examples are given.     

Symplectic reflections and complex space forms

We prove that a Hermitian manifold is a complex space form if and only if the local reflections with respect to any holomorphic surface are symplectic, i.e., preserve the Kähler form.     

Variational results on flag manifolds: Harmonic maps, geodesics and Einstein metrics

In this paper, we study variational aspects for harmonic maps from M to several types of flag manifolds and the relationship with the rich Hermitian geometry of these manifolds. We consider maps that are harmonic with respect to any invariant metric on each flag manifold. They are called equiharmonic maps. We survey some recent results for the case where M is a Riemann surface or is one dimensional; i.e., we study equigeodesics on several types of flag manifolds. We also discuss some results concerning Einstein metrics on such manifolds.     

On vanishing theorems for Higgs bundles

We introduce the notion of Hermitian Higgs bundle as a natural generalization of the notion of Hermitian vector bundle and we study some vanishing theorems concerning Hermitian Higgs bundles when the base manifold is a compact complex manifold. We show that a first vanishing result, proved for these objects when the base manifold was Kähler, also holds when the manifold is compact complex. From this fact and some basic properties of Hermitian Higgs bundles, we conclude several results. In particular we show that, in analogy to the classical case, there are vanishing theorems for invariant sections of tensor products of Higgs bundles. Then, we prove that a Higgs bundle admits no nonzero invariant sections if there is a condition of negativity on the greatest eigenvalue of the Hitchin–Simpson mean curvature. Finally, we prove that the invariant sections of certain tensor products of a weak Hermitian–Yang–Mills Higgs bundle are all parallel in the classical sense.     

Almost hermitian manifolds and Osserman condition

Let(M, g, J) be an almost Hermitian manifold. In this paper we study holomorphically nonnegatively(δ,Δ)-pinched almost Hermitian manifolds. In [3] it was shown that for such Kahler manifolds a plane with maximal sectional curvature has to be a holomorphic plane(J-invariant). Here we generalize this result to arbitrary almost Hermitian manifolds with respect to the holomorphic curvature tensorH R and toRK-manifolds of a constant type λ(p). In the proof some estimates of the sectional curvature are established. The results obtained are used to characterize almost Hermitian manifolds of constant holomorphic sectional curvature (with respect to holomorphic and Riemannian curvature tensor) in terms of the eigenvalues of the Jacobi-type operators, i.e. to establish partial cases of the Osserman conjecture. Some examples are studied. The first author is partially supported by SFS, Project #04M03.     

Partial energies monotonicity and holomorphicity of Hermitian pluriharmonic maps

In this paper,we introduce the notion of Hermitian pluriharmonic maps from Hermitian manifold into K¨ahler manifold.Assuming the domain manifolds possess some special exhaustion functions and the vecotor field V=JMδJM satisfies some decay conditions,we use stress-energy tensors to establish some monotonicity formulas of partial energies of Hermitian pluriharmonic maps.These monotonicity inequalities enable us to derive some holomorphicity for these Hermitian pluriharmonic maps.     

Riemannian manifolds with structure groupG 2

Summary Riemannian manifolds with structure group G ₂ are 7-dimensional and have a distinguished 3-form. In this paper such manifolds are treated as analogues of almost Hermitian manifolds. Thus S ⁷ has structure group G ₂ just as S ⁶ is an almost Hermitian manifold. We study the covariant derivative of the fundamental 3-form as was done in [GH]for almost Hermitian manifolds.     

On holomorphic planes

Summary In a manifold, almost complex, Hermitian or Kaehlerian, we shall introduce the notion of the axiom of holomorphic planes and show that a ϕ-connection in such a manifold is given strong restrictions if it satisfies the axiom of holomorphic planes. The main results are formulated in Theorems4.1, 4.2, 5.3 and5.7     

On the Geometry of Lagrangian Submanifolds

We prove that a Lagrangian submanifold passes through each point of a symplectic manifold in the direction of arbitrary Lagrangian plane at this point. Generally speaking, such a Lagrangian submanifold is not unique; nevertheless, the set of all such submanifolds in Hermitian extension of a symplectic manifold of dimension greater than 4 for arbitrary initial data contains a totally geodesic submanifold (which we call the s-Lagrangian submanifold) iff this symplectic manifold is a complex space form. We show that each Lagrangian submanifold in a complex space form of holomorphic sectional curvature equal to c is a space of constant curvature c/4. We apply these results to the geometry of principal toroidal bundles.     

Very Ampleness of Line Bundles and Canonical Embedding of Coverings of Manifolds

Let L be an ample line bundle on a Kähler manifolds of nonpositive sectional curvature with K as the canonical line bundle. We give an estimate of m such that K+mL is very ample in terms of the injectivity radius. This implies that m can be chosen arbitrarily small once we go deep enough into a tower of covering of the manifold. The same argument gives an effective Kodaira Embedding Theorem for compact Kähler manifolds in terms of sectional curvature and the injectivity radius. In case of locally Hermitian symmetric space of noncompact type or if the sectional curvature is strictly negative, we prove that K itself is very ample on a large covering of the manifold.     

An anomaly formula for Ray–Singer metrics on manifolds with boundary

Using the heat kernel, we derive first a local Gauss–Bonnet–Chern theorem for manifolds with a non-product metric near the boundary. Then we establish an anomaly formula for Ray–Singer metrics defined by a Hermitian metric on a flat vector bundle over a Riemannian manifold with boundary, not assuming that the Hermitian metric on the flat vector bundle is flat nor that the Riemannian metric has product structure near the boundary. Received: January 2004; Revision: February 2005; Accepted: September 2005     

On almost hyperHermitian structures on Riemannian manifolds and tangent bundles

Some results concerning almost hyperHermitian structures are considered, using the notions of the canonical connection and the second fundamental tensor field h of a structure on a Riemannian manifold which were introduced by the second author. With the help of any metric connection on an almost Hermitian manifold M an almost hyperHermitian structure can be constructed in the defined way on the tangent bundle TM. A similar construction was considered in [6], [7]. This structure includes two basic anticommutative almost Hermitian structures for which the second fundamental tensor fields h ¹ and h ² are computed. It allows us to consider various classes of almost hyperHermitian structures on TM. In particular, there exists an infinite-dimensional set of almost hyperHermitian structures on TTM where M is any Riemannian manifold.     

Rotations and Hermitian symmetric spaces

On an almost Hermitian manifold (M, g, J) one considers the naturally defined field of local diffeomorphismsj _m =exp _m J _m exp _m ^–1 ,mM, and in particular, one studies isometric, harmonic, holomorphic and symplecticj _m . This leads to some characterizations of special classes of almost Hermitian manifolds, including the class of Hermitian symmetric spaces. In addition, one treats some intrinsic and extrinsic geometrical properties of geodesic spheres relating to these local diffeomorphisms.Supported by grant 203.01.50 of the C.N.R., Italy.     

Reflection groups on Riemannian manifolds

We investigate discrete groups G of isometries of a complete connected Riemannian manifold M which are generated by reflections, in particular those generated by disecting reflections. We show that these are Coxeter groups, and that the orbit space M/G is isometric to a Weyl chamber C which is a Riemannian manifold with corners and certain angle conditions along intersections of faces. We can also reconstruct the manifold and its action from the Riemannian chamber and its equipment of isotropy group data along the faces. We also discuss these results from the point of view of Riemannian orbifolds. Mathematics Subject Classification Primary 51F15, 53C20, 20F55, 22E40     

Polar actions on Hermitian and quaternion-Kähler symmetric spaces

We analyze polar actions on Hermitian and quaternion-Kähler symmetric spaces of compact type. For complex integrable polar actions on Hermitian symmetric spaces of compact type we prove a reduction theorem and several corollaries concerning the geometry of these actions. The results are independent of the classification of polar actions on Hermitian symmetric spaces. In the second part we prove that polar actions on Wolf spaces are quaternion-coisotropic and that isometric actions on these spaces admit an orbit of special type, analogous to the existence of a complex orbit for an isometric action on a compact homogeneous simply connected Kähler manifold.     

Canonical connection on a class of Reimannian almost product manifolds

The canonical connection on a Reimannian almost product manifold is an analogue to the Hermitian connection on an almost Hermitian manifold. In this paper we consider the canonical connection on a class of Reimannian almost product manifolds with non-integrable almost product structure.     

On Einstein Hermitian manifolds

We show that every compact Einstein Hermitian surface with constant *–scalar curvature is a K?hler surface. In contrast to the 4-dimensional case, it is shown that there exists a compact Einstein Hermitian (4n + 2)-dimensional manifold with constant *–scalar curvature which is not K?hler. This study is supported by Kangwon National University.     

Laplacians on the holomorphic tangent bundle of a Kaehler manifold

Let M be a connected complex manifold endowed with a Hermitian metric g. In this paper, the complex horizontal and vertical Laplacians associated with the induced Hermitian metric 〈·, ·〉 on the holomorphic tangent bundle T ^1,0 M of M are defined, and their explicit expressions are obtained. Using the complex horizontal and vertical Laplacians associated with the Hermitian metric 〈·, ·〉 on T ^1,0 M, we obtain a vanishing theorem of holomorphic horizontal p forms which are compactly supported in T ^1,0 M under the condition that g is a Kaehler metric on M.     

A class of locally $\varphi $-symmetric contact metric spaces

We prove that a non-Sasakian contact metric manifold such that its characteristic vector field belongs to the (k,m)(k,\mu )-nullity distribution, is locally homogeneous. Further, these spaces are all locally j\varphi -symmetric in the strong sense, i.e., the reflections with respect to characteristic curves are local isometries.