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Pseudo-spherical submanifolds with 1-type pseudo-spherical Gauss map
In this work, we study the pseudo-Riemannian submanifolds of a pseudo-sphere
with 1-type pseudo-spherical Gauss map. First, we classify the Lorentzian
surfaces in a 4-dimensional pseudo-sphere with index s,
, and having harmonic pseudo-spherical Gauss map. Then we give a
characterization theorem for pseudo-Riemannian submanifolds of a pseudo-sphere
with 1-type pseudo-spherical Gauss
map, and we classify spacelike surfaces and Lorentzian surfaces in the de
Sitter space with 1-type
pseudo-spherical Gauss map. Finally, according to the causal character of the
mean curvature vector we obtain the classification of submanifolds of a
pseudo-sphere having 1-type pseudo-spherical Gauss map with nonzero constant
component in its spectral decomposition
Pseudo-dualizing complexes and pseudo-derived categories
The definition of a pseudo-dualizing complex is obtained from that of a
dualizing complex by dropping the injective dimension condition, while
retaining the finite generatedness and homothety isomorphism conditions. In the
specific setting of a pair of associative rings, we show that the datum of a
pseudo-dualizing complex induces a triangulated equivalence between a
pseudo-coderived category and a pseudo-contraderived category. The latter terms
mean triangulated categories standing "in between" the conventional derived
category and the coderived or the contraderived category. The constructions of
these triangulated categories use appropriate versions of the Auslander and
Bass classes of modules. The constructions of derived functors providing the
triangulated equivalence are based on a generalization of a technique developed
in our previous paper arXiv:1503.05523.Comment: LaTeX 2e with pb-diagram, xy-pic, and tikz-cd, 60 pages, 14+3
commutative diagrams; v.4: sections 10-12 added, new subsections 0.8 and 0.10
inserted in the Introduction; v.5: subsection 0.2 shortened, basic
definitions added in section 1, explanations added in the proof of Theorem
4.2, several references added; v.7: misprints corrected, references updated
-- this is intended as the final versio
Covariant derivative of the curvature tensor of pseudo-K\"ahlerian manifolds
It is well known that the curvature tensor of a pseudo-Riemannian manifold
can be decomposed with respect to the pseudo-orthogonal group into the sum of
the Weyl conformal curvature tensor, the traceless part of the Ricci tensor and
of the scalar curvature. A similar decomposition with respect to the
pseudo-unitary group exists on a pseudo-K\"ahlerian manifold; instead of the
Weyl tensor one obtains the Bochner tensor. In the present paper, the known
decomposition with respect to the pseudo-orthogonal group of the covariant
derivative of the curvature tensor of a pseudo-Riemannian manifold is refined.
A decomposition with respect to the pseudo-unitary group of the covariant
derivative of the curvature tensor for pseudo-K\"ahlerian manifolds is
obtained. This defines natural classes of spaces generalizing locally symmetric
spaces and Einstein spaces. It is shown that the values of the covariant
derivative of the curvature tensor for a non-locally symmetric
pseudo-Riemannian manifold with an irreducible connected holonomy group
different from the pseudo-orthogonal and pseudo-unitary groups belong to an
irreducible module of the holonomy group.Comment: the final version accepted to Annals of Global Analysis and Geometr
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