A Splitting Result for Real Submanifolds of a Kähler Manifold

2021

Let (Z, ω) be a Kähler manifold and let U be a compact connected Lie group with Lie algebra u acting on Z and preserving ω. We assume that the U -action extends holomorphically to an action of the complexified group U and the U -action on Z is Hamiltonian. Then there exists a U -equivariant momentum map μ : Z → u. If G ⊂ U is a closed subgroup such that the Cartan decomposition U = Uexp(iu) induces a Cartan decomposition G = Kexp(p), where K = U ∩ G, p = g ∩ iu and g = k ⊕ p is the Lie algebra of G, there is a corresponding gradient map μp : Z → p. If X is a G-invariant compact and connected real submanifold of Z, we may consider μp as a mapping μp : X → p. Given an Ad(K)-invariant scalar product on p, we obtain a Morse like function f = 1 2 ‖ μp ‖ 2 on X. We point out that, without the assumption that X is real analytic manifold, the Lojasiewicz gradient inequality holds for f . Therefore the limit of the negative gradient flow of f exists and it is unique. Moreover, we prove that ...

Transactions of the American Mathematical Society, 1974

Complex analytic submanifolds and totally real submanifolds are two typical c l a s s e s among all submanifolds of an almost Hermitian manifold. In this paper, some characterizations of totally real submanifolds are given. Moreover some classifications of totally real submanifolds in complex space forms are obtained. (resp. M). We call M an totally real suJbmanifold of M if M admits an isometric CU immersion into M such that for all x, ] (Tx(M))C v X , where Tx(M)denotes the tangent space of M at x and vx the normal space at x . By a plane section we mean a 2-dimensional linear subspace of a tangent CU space. A plane section r i s called antiholomorphic if ]r i s perpendicular to 7. Proposition 2.1. Let M be a submanifold immersed in an almost Hermitian CU .-u manifold M. T h e n M i s a totally real submanifold of M if and only if every plane section of M is antiholomorphic. Proof. Let X be an arbitrary vector in Tx(M),and let e l = X I e 2 , .*, en be a basis of Tx(M). We denote by rij the plane section spannei by ei and e j . Assume that every plane section i s antiJolomorphic. Then ] r l . are perpen-

Monatshefte f�r Mathematik, 1981

Let N be a real submanifold in a complex manifold M. If the maximal complex subspaces of the tangent spaces of M contained in the tangent spaces of N are of constant dimension and they define a differentiable distribution, then N is called a generic submanifold. The class of generic submanifold includes all real hypersurfaces, complex submanifolds, totally real submanifolds and CR-submanifolds. In this paper we initiate a study of generic submanifolds in a K~hler manifold from differential geometric point of view. Some fundamental results in this respect will be obtained.

Journal of Differential Geometry, 1977

Asian Journal of Mathematics, 2004

We prove geodesic completeness and global conectivity for a step 2k + 2 subRiemannian manifold. Using complex Hamiltonian mechanics we also calculate some subRiemannian distances. *

The Journal of Geometric Analysis

We presented a systematic treatment of a Hilbert criterion for stability theory for an action of a real reductive group G on a real submanifold X of a Kähler manifold Z. More precisely, we suppose that the action of a compact Lie group with Lie algebra u extends holomorphically to an action of the complexified group U C and that the U-action on Z is Hamiltonian. If G ⊂ U C is compatible, there is a corresponding gradient map µp : X → p, where g = k ⊕ p is a Cartan decomposition of the Lie algebra of G. The concept of energy complete action of G on X is introduced. For such actions, one can characterize stability, semistability and polystability of a point by a numerical criteria using a G-equivariant function called maximal weight. We also prove the classical Hilbert-Mumford criteria for semistability and polystability conditions.

Journal of the Australian Mathematical Society, 1998

Recently, Chen defined an invariant δM of a Riemannian manifold M. Sharp inequalities for this Riemannian invariant were obtained for submanifolds in real, complex and Sasakian space forms, in terms of their mean curvature. In the present paper, we investigate certain C-totally real submanifolds of a Sasakian space form M2m+1(C)satisfying Chen's equality.

Mathematische Zeitschrift, 1988

Arab Journal of Mathematical Sciences, 2014

Inannian manifold isometrically unnierseci in _ll. \Ve note that subnlanifolds of a Kaehler manifold are determined by the behaviour of tali , nt bundle of the sul>nlanifold under the action of t he almost a1111ple1 structure of the ambient. miianifokl. A si1hnhanifol(l ;1I is called holoIUorl)llic (coulplex) if J(1 ,(,11)) C 1,(:11) for revery p E Al. where 7(M) (leuotes the tangent space to .11 at the point p. .11 is called totally real if .I (7(.11)) C 71; (.11), for every p E .11, where 1 r (.\I) denotes the normal space to .11 at the point p. As a generalization of holomnorpilic and totally real submamiifolds. CI?-suhnlanifolds were introduced by A. Be.lanco. A C11-sIm1nlallifold .11 of an almost Hermitian manifold ,1/ with an almost complex striIetilre J requires tWWWu, orthogonal couhlllenmelItry distributions D and DL defined on .11 such that D is invariant under .1 and Dl is totally real (A. Bejaneu. 1978: B. Y. ('hen. 1981a). There is yet another generalization of ('R-sulnnanifolds known as generic tiupl1ianifok1s (B. Y. Chen. 1981c). These snbnnahifolds are defined by relaxing the condition on the coinp1enlentary distribution of ho1olhorlhic distribution. Let .\I be a real subilianifold of all alms t Hermit ian manifold .1I and let DI, = TI,:'lt tl .JTI .lI he he iimaximal hololnorphic suhspace of '11,(:11). If D : p-V t, definesa sinooth twloulorpliie distribution on .11. then .11 is called a generic SilbI11aIllfOl(l of M. The complementary distribution D" of V is called purely real distribution on .11. A generic sobxuanifold is a CR-subJualifcl(1 if the purely real distribution on .11 is totally real. A purely real distribution D" on a generic subinanifohi Al is called proper if it is not totally real. A generic subnlanifoki is called proper if purely real c1istribittiou is proper. Iiaehler manifold and obtained the relations between holomorphic sectional curvatures of Al restricted to D and that of B. Further this study has been extended by S. Deshmukh, S. Ali and S. 1. Husain (1988). in which they obtained the relations between the Ricci curvatures and the scalar curvatures of a Kaehler manifold and the base manifold. To deal with the similar question for the generic submanifold of a Kaehler rnanifold, one has the difficulty that the distribution V' for generic simbntauifold of it Kaehler manifold is not necessarily integrable to match the requirement of the submersion. To overcome this difficulty we consider the submersion r, :.11-> B of generic submanifolds Al of a Kaehler manifold Al onto an almost Hermitian manifold B with the assumption that. D" is integrable. In the first. section of this chapter. we have considered submersion of a Cfl subinanifold of it Kaehler manifold l onto an almost Hermitian manifold and obtain product, theorems on such submersion by imposing conditions on CR-submanifold and its distribution. In section 2 we study the submersions of generic submanifolds Al of a Kaehler manifold Al onto an almost Hermitian manifold B with integrable purely rail distribution D° and Drove that for the submersion of a generic submnanifold A! of a Kaeliler manifold Al onto an almost Hermitian manifold B. B is necessarily a Kaeliler manifold and obtain time decomposition theoremms for the generic submnanifold Al. Also we 2.1. Submersion of CR-Siihmanifolds 21 have obtained the relation between the holomorphic sectional curvatures of :1f1 restricted to D and that. of B. In the last of this chapter we have obtained the relation between the holornorphic sectional curvatures of :11 restricted to D and that of B with totally geodesic fibres. The contents of this chapter are partially published in Global Journal of Advanced Research on Classical and Modern Geometries. 2.1 Submersion of C Yf-Submanifolds In this section. we have studied the. CR-suhmersiutis of lKaehlcr manifolds and a tnunber of decomposition t.heoreins have been proved here. We start this section with the definition of the stibinersion of a CJ?-subiiianifold of an almost. Hcrmit.ian manifold.

Let $f\colon M^{2n}\to\R^{2n+p}$, $2\leq p\leq n-1$, be an isometric immersion of a Kaehler manifold into Euclidean space. Yan and Zheng conjectured in \cite{YZ} that if the codimension is $p\leq 11$ then, along any connected component of an open dense subset of $M^{2n}$, the submanifold is as follows: it is either foliated by holomorphic submanifolds of dimension at least $2n-2p$ with tangent spaces inthe kernel of the second fundamental form whose images are open subsets of affine vector subspaces, or it is embedded holomorphically in a Kaehler submanifold of $\R^{2n+p}$ of larger dimension than $2n$. This bold conjecture was proved by Dajczer and Gromoll just for codimension three and then by Yan and Zheng for codimension four. In this paper we prove that the second fundamental form of the submanifold behaves pointwise as expected in case that the conjecture is true. This result is a first fundamental step fora possible classification of the non-holomorphic Kaehler submanifolds l...