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1.
We prove a Berger type theorem for the normal holonomy F^{\Phi^\perp} (i.e., the holonomy group of the normal connection) of a full complete complex submanifold M of the complex projective space
\mathbbC Pn{\mathbb{C} P^n}. Namely, if F^{\Phi^\perp} does not act transitively, then M is the complex orbit, in the complex projective space, of the isotropy representation of an irreducible Hermitian symmetric
space of rank greater or equal to 3. Moreover, we show that for complete irreducible complex submanifolds of
\mathbbCn{\mathbb{C}^n} the normal holonomy is generic, i.e., it acts transitively on the unit sphere of the normal space. The methods in the proofs
rely heavily on the singular data of appropriate holonomy tubes (after lifting the submanifold to the complex Euclidean space,
in the
\mathbbC Pn{\mathbb{C} P^n} case) and basic facts of complex submanifolds. 相似文献
2.
From the existence of parallel spinor fields on Calabi-Yau, hyper-Kähler or complex flat manifolds, we deduce the existence of harmonic differential forms of different degrees on their minimal Lagrangian submanifolds. In particular, when the submanifolds are compact, we obtain sharp estimates on their Betti numbers which generalize those obtained by Smoczyk in [49]. When the ambient manifold is Kähler-Einstein with positive scalar curvature, and especially if it is a complex contact manifold or the complex projective space, we prove the existence of Kählerian Killing spinor fields for some particular spin c structures. Using these fields, we construct eigenforms for the Hodge Laplacian on certain minimal Lagrangian submanifolds and give some estimates for their spectra. These results also generalize some theorems by Smoczyk in [50]. Finally, applications on the Morse index of minimal Lagrangian submanifolds are obtained. 相似文献
3.
Toru Sasahara 《Czechoslovak Mathematical Journal》2014,64(1):79-90
An explicit representation for ideal CR submanifolds of a complex hyperbolic space has been derived in T. Sasahara (2002). We simplify and reformulate the representation in terms of certain Kähler submanifolds. In addition, we investigate the almost contact metric structure of ideal CR submanifolds in a complex hyperbolic space. Moreover, we obtain a codimension reduction theorem for ideal CR submanifolds in a complex projective space. 相似文献
4.
5.
Considering the Levi form on CR submanifolds of maximal CR dimension of complex space forms, we prove that on some remarkable
real submanifolds of complex projective space the Levi form can never vanish and we determine all such submanifolds in the
case when the ambient manifold is a complex Euclidean space.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
6.
In this paper, we investigate complete curvature-adapted submanifolds with maximal flat section and trivial normal holonomy group in symmetric spaces of compact type or non-compact type under a certain condition, and derive the constancy of the principal curvatures of such submanifolds. As a result, we derive that such submanifolds are isoparametric. 相似文献
7.
In this paper we show how the restriction of the complex algebraic cycles to real part of a complex algebraic set is related
to the real algebraic cycles of the real part. As a corollary we give examples of smooth submanifolds of a Euclidean space
which can not be isotoped to real parts of complex nonsingular subvarieties in the corresponding projective space.
Dedicated to memory of Mario Raimondo 相似文献
8.
证明了复射影空间中两种类型法丛平坦的全实迷向予流形必是极小的,并在紧致的情形确定了它们的具体形状. 相似文献
9.
Summary The ideas of holonomy group fixing an m-dimensional plane in a Finsler space were given by one of the present authors[1]. In that paper the deformation properties of the space admitting such holonomy group were of main consideration and, indeed,
the decomposition characteristics of the space were not touched upon. In the present paper we consider the decomposition of
the space due to the existence of holonomy group. The geometry is constructed on the decomposed metric of the space. The decomposition
properties of various entities such as the connection parameters, the covariant derivatives, the curvature tensors, and the
projective curvature tensors have been studied. In all there are six articles in the paper. The first of these is introductory.
The next three articles are dealt with the Cartan's approach to Finsler space whereas the fifth one is dealt with Berwald's
approach. The last article is devoted to the theory of decomposition in the projective curvature tensors.
Entrata in Redazione il 18 ottobre 1969. 相似文献
10.
Quo-Shin Chi 《Annals of Global Analysis and Geometry》1991,9(3):197-204
Using the twistor theory on quaternionic Kaehler manifolds and some recent results on Blaschke manifolds and compact manifolds whose holonomy group is Spin (7), we prove that a Blaschke manifold of nonnegative scalar curvature whose holonomy group is exceptional is isometric to a projective space. 相似文献
11.
Alekseevsky Dmitri V.; Di Scala Antonio J. 《Proceedings London Mathematical Society》2004,89(1):193-216
We study the (restricted) holonomy group Hol() of the normalconnection (shortened to normal holonomy group) of a Kählersubmanifold of a complex space form. We prove that if the normalholonomy group acts irreducibly on the normal space then itis linear isomorphic to the holonomy group of an irreducibleHermitian symmetric space. In particular, it is a compact groupand the complex structure J belongs to its Lie algebra. We prove that the normal holonomy group acts irreducibly ifthe submanifold is full (that is, it is not contained in a totallygeodesic proper Kähler submanifold) and the second fundamentalform at some point has no kernel. For example, a KählerEinsteinsubmanifold of CPn has this property. We define a new invariant µ of a Kähler submanifoldof a complex space form. For non-full submanifolds, the invariantµ measures the deviation of J from belonging to the normalholonomy algebra. For a KählerEinstein submanifold,the invariant µ is a rational function of the Einsteinconstant. By using the invariant µ, we prove that thenormal holonomy group of a not necessarily full KählerEinsteinsubmanifold of CPn is compact, and we give a list of possibleholonomy groups. The approach is based on a definition of the holonomy algebrahol(P) of an arbitrary curvature tensor field P on a vectorbundle with a connection and on a De Rham type decompositiontheorem for hol(P). 2000 Mathematics Subject Classification53C40 (primary), 53B25 (secondary). 相似文献
12.
In this paper we show that Hamiltonian stable minimal Lagrangian submanifolds of projective space need not have parallel second
fundamental form. 相似文献
13.
本文研究了复射影空间中的全实2 -调和子流形问题.利用活动标架法,获得了这类子流形成为极小子流形的关于第二基本形式模长的Pinching定理及一个积分不等式.此外还得到关于全实2-调和伪脐子流形的一些刚性定理,推广了CPn中全实2-调和子流形的一些相应结果. 相似文献
14.
We study the Ribaucour transformation for flat Lagrangian submanifolds in complex flat space and complex projective space.
As a consequence, we obtain a process to generate a new family of such submanifolds from a given one. Analytically, this provides
a method to construct new solutions of the corresponding systems of PDEs from a given one. We also show that such transformation
always comes with a permutability formula. 相似文献
15.
16.
We consider immersions: and construct a subspace of which corresponds to a set of embedded manifolds which are either parallel to f, tubes around f or, in general, partial tubes around f. This space is invariant under the action of the normal holonomy group, We investigate the case where is non-trivial and obtain some results on the number of connected components of .
Received 24 March 2000. 相似文献
17.
18.
We study compact minimal generic submanifolds of a complex projective space with flat normal connection and prove a reduction theorem of codimension under the condition on the Ricci tensor.The present studies were supported by the Basic Science Research Institute Program, Korea Ministry of Education, 1993-114. 相似文献
19.
The planar geodesic submanifolds of a quaternionic projective space are studied. Especially, these submanifolds which are totally real or quaternionic CR-submanifolds are completely classified. Also, the non-existence of a planar geodesic, proper QR-product in a quaternionic projective space is proved.Research supported in part by a grant from KOSEF. 相似文献
20.
Pablo Alegre Bang-Yen Chen Marian Ioan Munteanu 《Annals of Global Analysis and Geometry》2012,42(3):317-331
In this study, we establish a sharp relation between δ-invariants and Riemannian submersions with totally geodesic fibers. By using this relationship, we establish an optimal inequality involving δ-invariants for submanifolds of the complex projective space CP m (4) via Hopf’s fibration ${\pi:S^{2m+1}\to CP^{m}(4)}$ . Moreover, we completely classify submanifolds of complex projective space which satisfy the equality case of the inequality. 相似文献