The main two families of real hypersurfaces in complex space forms are Hopf and ruled. However, very little is known about the indefinite projective \({\mathbb {C}}P^n_p\). In a previous work, Kimura second author introduced this paper, ruled introduced, as those whose maximal holomorphic distribution integrable, such that leaves totally geodesic hyperplanes. A detailed description shape operat...

p. (x) l. 5, “Chapter 4” should read “Chapter 6”. p. 24–25: As it stands, the proof of Lemma 2.2.3 is wrong, because ψ is not necessarily surjective. However, if P is a finitely generated projective module, then the proof works. So at the beginning of the section, one should make the further observation that P = F ′(ΓΓ) is a finitely generated projective module. This is because among all projec...

For a weighted projective line, the stable category of its vector bundles modulo line bundles has a natural triangulated structure. We prove that, for any positive integers p, q, r and r′ with r′ r, there is an explicit recollement of the stable category of vector bundles on a weighted projective line of weight type (p, q, r) relative to the ones on weighted projective lines of weight types (p,...

On a manifold with a projective connection we canonically assign a second order differential operator acting on the algebra of all densities to any tensor density S of fixed weight λ. In particular, this implies that on any projectively connected manifold, a ‘bracket’ (symmetric biderivation) on the algebra of functions extends canonically to the algebra of densities.

In this paper, we describe $ss$-supplement submodules in terms of a special class endomorphisms. Let $R$ be ring with semisimple radical and $P$ projective $R-$module. We show that there is bijection between ss-supplement $End_{R}(P)$. Moreover, define radical-s-projective modules as generalization modules. prove every submodule $R-$module over the radical. $SSI$-ring $R$, projective. provide r...

Abstract We show that compact Kähler manifolds have the rational cohomology ring of complex projective space provided a weighted sum lowest three eigenvalues curvature operator is positive. This follows from more general vanishing and estimation theorem for individual Hodge numbers. also prove an analogue Tachibana’s manifolds.