Real hypersurfaces in complex projective space whose structure Jacobi operator is Lie ξ-parallel

We classify real hypersurfaces in complex projective spaces whose structure Jacobi operator is Lie parallel in the direction of the structure vector field.     

Naturally reductive Riemannian homogeneous structures on some classes of generic submanifolds in complex space forms

We prove that the universal covering spaces of the generic submanifolds of C P _n and of C H _n are naturally reductive homogeneous spaces by determining explicitly tensor fields defining naturally reductive homogeneous structures on them.     

A characterization of constant isotropic immersions by circles

Motivated by the well-known result of Nomizu and Yano [4], we provide a characterization of constant isotropic immersions into an arbitrary Riemannian manifold by circles on the submanifolds. As an immediate consequence of this result, we characterize Veronese imbeddings of complex projective spaces into complex projective spaces which are typical examples of Kähler immersions. Received: 11 January 2002     

Totally geodesic submanifolds of the complex quadric

In this article, relations between the root space decomposition of a Riemannian symmetric space of compact type and the root space decompositions of its totally geodesic submanifolds (symmetric subspaces) are described. These relations provide an approach to the classification of totally geodesic submanifolds in Riemannian symmetric spaces; this is exemplified by the classification of the totally geodesic submanifolds in the complex quadric Qm:=SO(m+2)/(SO(2)×SO(m)) obtained in the second part of the article. The classification shows that the earlier classification of totally geodesic submanifolds of Qm by Chen and Nagano (see [B.-Y. Chen, T. Nagano, Totally geodesic submanifolds of symmetric spaces, I, Duke Math. J. 44 (1977) 745-755]) is incomplete. More specifically, two types of totally geodesic submanifolds of Qm are missing from [B.-Y. Chen, T. Nagano, Totally geodesic submanifolds of symmetric spaces, I, Duke Math. J. 44 (1977) 745-755]: The first type is constituted by manifolds isometric to CP¹×RP¹; their existence follows from the fact that Q² is (via the Segre embedding) holomorphically isometric to CP¹×CP¹. The second type consists of 2-spheres of radius which are neither complex nor totally real in Qm.     

Symplectic, complex and Kähler structures on four-dimensional generalized symmetric spaces

We obtain the full classification of invariant symplectic, (almost) complex and Kähler structures, together with their paracomplex analogues, on four-dimensional pseudo-Riemannian generalized symmetric spaces. We also apply these results to build some new examples of five-dimensional homogeneous K-contact, Sasakian, K-paracontact and para-Sasakian manifolds.     

Einstein condition and twistor spaces of compatible partially complex structures

We study the Einstein condition for a natural family of Riemannian metrics on the twistor space of partially complex structures of a fixed rank on the tangent spaces of a Riemannian manifold compatible with its metric. A generalization of the Einstein condition (discussed in the Besse book [Enstein Manifolds, Ergeb. Math. Grensgeb. (3), vol. 10, Springer, New York, 1987]) is also considered.     

A splitting theorem for compact complex homogeneous spaces with a symplectic structure

In this note we prove a splitting theorem for compact complex homogeneous spaces with a cohomology 2 class [] such that the top power [ ⁿ ]0.Dedicated to Professor W. C. Hsiang on the occasion of his 60th birthdayPartially supported by NSF Grant DMS-9401755.     

Six-dimensional product Lie algebras admitting integrable complex structures

We classify the 6-dimensional Lie algebras of the form $\mathfrak{g}\times \mathfrak{g}$ that admit an integrable complex structure. We also endow a Lie algebra of the kind $\mathfrak{o}(n)\times \mathfrak{o}(n)$ ($n\ge 2$) with such a complex structure. The motivation comes from geometric structures à la Sasaki on $\mathfrak{g}$-manifolds.     

Geometry of Warped Product CR-Submanifolds in Kaehler Manifolds, II

 A CR-submanifold N of a Kaehler manifold is called a CR-warped product if N is the warped product of a holomorphic submanifold and a totally real submanifold of . This notion of CR-warped products was introduced in part I of this series. It was proved in part I that every CR-warped product in a Kaehler manifold satisfies a basic inequality: . The classification of CR-warped products in complex Euclidean space satisfying the equality case of the inequality is archived in part I. The main purpose of this second part of this series is to classify CR-warped products in complex projective and complex hyperbolic spaces which satisfy the equality. (Received 13 March 2001; in revised form 10 August 2001)     

On the geometry of complex ‐spaces

It is known that applying an ‐homothetic deformation to a complex contact manifold whose vertical space is annihilated by the curvature yields a condition which is invariant under ‐homothetic deformations. A complex contact manifold satisfying this condition is said to be a complex ‐space. In this paper, we deal with the questions of Bochner, conformal and conharmonic flatness of complex ‐spaces when , and prove that such kind of spaces cannot be Bochner flat, conformally flat or conharmonically flat.     

The scalar curvature of CR submanifolds of maximal CR dimension of complex projective space

We treat n-dimensional compact minimal submanifolds of complex projective space when the maximal holomorphic tangent subspace is (n − 1)-dimensional and we give a sufficient condition for such submanifolds to be tubes over totally geodesic complex subspaces. Authors’ addresses: Mirjana Djorić, Faculty of Mathematics, University of Belgrade, Studentski trg 16, pb. 550, 11000 Belgrade, Serbia; Masafumi Okumura, 5-25-25 Minami Ikuta, Tama-ku, Kawasaki, Japan     

Gauss Equation And Injectivity Radii For Subspaces in Spaces of Curvature Bounded Above

A Gauss Equation is proved for subspaces of Alexandrov spaces of curvature bounded above by K. That is, a subspace of extrinsic curvature ⩽ A, defined by a cubic inequality on the difference of arc and chord, has intrinsic curvature ⩽ K+A². Sharp bounds on injectivity radii of subspaces, new even in the Riemannian case, are derived.     

Skew-symmetric derivations of a complex Lie algebra

Let be a complex Lie algebra, its underlying real Lie algebra, a real form of and ·, · the euclidean product induced by the real part of an hermitian inner product on . Let aut be the Lie algebra of skew-symmetric derivations of . We give necessary and sufficient conditions to ensure that aut is composed of skew-hermitian derivations. As an application, we study holomorphy in large subgroups of isometries of Lie groups.     

The classification of 4-dimensional homogeneous D'Atri spaces revisited

In this short note we correct the (incomplete) classification theorem from [F. Podestà, A. Spiro, Four-dimensional Einstein-like manifolds and curvature homogeneity, Geom. Dedicata 54 (1995) 225-243], we improve a result from [P. Bueken, L. Vanhecke, Three- and four-dimensional Einstein-like manifolds and homogeneity, Geom. Dedicata 75 (1999) 123-136] and we announce the final solution of the classification problem for 4-dimensional homogeneous D'Atri spaces.     

Minimal 2-spheres in a complex projective space

In this paper conformal minimal 2-spheres immersed in a complex projective space are studied by applying Lie theory and moving frames. We give differential equations of Kähler angle and square length of the second fundamental form. By applying these differential equations we give characteristics of conformal minimal 2-spheres of constant Kähler angle and obtain pinching theorems for curvature. We also discuss conformal minimal 2-spheres of constant normal curvature and prove that there does not exist any linearly full minimal 2-sphere immersed in a complex projective space CPn (n>2) with non-positive constant normal curvature. We also prove that a linearly full minimal 2-sphere immersed in a complex projective space CPn (n>2) with constant normal curvature and constant Kähler angle is of constant curvature.     

Special isoparametric orbits in Riemannian symmetric spaces

In the present paper orbits of isotropy subgroups in Riemannian symmetric spaces are discussed. Principal orbits of an isotropy subgroup are isoparametric in the sense of Palais and Terng (seeCritical Point Theory and Submanifold Geometry, Springer-Verlag, Berlin, 1988). We show that excepting some special cases, the shape operator with respect to the radial unit vector field determines a totally geodesic foliation on a given principal orbit. Furthermore, we prove that the shape operators and the curvature endomorphisms with respect to the normal vectors commute on these isoparametric submanifolds.     

On real and complex Berwald connections associated to strongly convex weakly Kähler-Finsler metric

In this paper, we give a definition of weakly complex Berwald metric and prove that, (i) a strongly convex weakly Kähler-Finsler metric F on a complex manifold M is a weakly complex Berwald metric iff F is a real Berwald metric; (ii) assume that a strongly convex weakly Kähler-Finsler metric F is a weakly complex Berwald metric, then the associated real and complex Berwald connections coincide iff a suitable contraction of the curvature components of type (2,0) of the complex Berwald connection vanish; (iii) the complex Wrona metric in Cn is a fundamental example of weakly complex Berwald metric whose holomorphic curvature and Ricci scalar curvature vanish identically. Moreover, the real geodesic of the complex Wrona metric on the Euclidean sphere S2n−1⊂Cn is explicitly obtained.     

Quaternionic analytic torsion

We define an (equivariant) quaternionic analytic torsion for anti-self-dual vector bundles on quaternionic Kähler manifolds, using ideas by Leung and Yi. We do so by constructing a Laplace operator associated to a complex defined by Salamon. We compute this torsion for vector bundles on quaternionic homogeneous spaces with respect to any isometry in the component of the identity, in terms of roots and Weyl groups.     

Uniqueness of complex structure and real hereditarily indecomposable Banach spaces

There exists a real hereditarily indecomposable Banach space X=X(C) (respectively X=X(H)) such that the algebra L(X)/S(X) is isomorphic to C (respectively to the quaternionic division algebra H).Up to isomorphism, X(C) has exactly two complex structures, which are conjugate, totally incomparable, and both hereditarily indecomposable. So there exist two Banach spaces which are isometric as real spaces but totally incomparable as complex spaces. This extends results of J. Bourgain and S. Szarek [J. Bourgain, Real isomorphic complex Banach spaces need not be complex isomorphic, Proc. Amer. Math. Soc. 96 (2) (1986) 221-226; S. Szarek, On the existence and uniqueness of complex structure and spaces with “few” operators, Trans. Amer. Math. Soc. 293 (1) (1986) 339-353; S. Szarek, A superreflexive Banach space which does not admit complex structure, Proc. Amer. Math. Soc. 97 (3) (1986) 437-444], and proves that a theorem of G. Godefroy and N.J. Kalton [G. Godefroy, N.J. Kalton, Lipschitz-free Banach spaces, Studia Math. 159 (1) (2003) 121-141] about isometric embeddings of separable real Banach spaces does not extend to the complex case.The quaternionic example X(H), on the other hand, has unique complex structure up to isomorphism; other examples with a unique complex structure are produced, including a space with an unconditional basis and non-isomorphic to l₂. This answers a question of S. Szarek in [S. Szarek, A superreflexive Banach space which does not admit complex structure, Proc. Amer. Math. Soc. 97 (3) (1986) 437-444].