We study quotients of multi-graded bundles, including double vector bundles. Among other things, we show that any such quotient fits into a tower affine Applications the theory include construction normal bundles for weighted submanifolds, as well pairs submanifolds with clean intersection.

We solve the generalized Milnor–Hirzebruch problem on the relations between Chern numbers of an arbitrary stably almost complex manifold and Chern numbers of its virtual Chern submanifolds. As a corollary we derive new divisibility conditions for Chern classes of an arbitrary complex vector bundle over a stably almost complex manifold. Introduction The generalized Milnor–Hirzebruch problem was ...

From the existence of parallel spinor fields on CalabiYau, 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. When the ambient manifold is Kähler-Einstein with positive scalar curvature, and...

A second order family of special Lagrangian submanifolds of C m is a family characterized by the satisfaction of a set of pointwise conditions on the second fundamental form. For example, the set of ruled special Lagrangian submanifolds of C 3 is characterized by a single algebraic equation on the second fundamental form. While the ‘generic’ set of such conditions turns out to be incompatible, ...

The theory of finite type submanifolds was introduced by the first author in late 1970s and it has become a useful tool for investigation submanifolds. Later, P. Piccinni extended notion to maps submanifolds; particular, with Gauss map. Since then, there have been rapid developments type. simplest are those which 1-type. classes such constitute very special interesting families theory.

We introduce the geometry of pseudo-totally umbilical lightlike submanifold $M$ a semi-Riemannian manifold $\bar{M}$. In line with above, we give complete classification $1$-lightlike submanifolds, such as hypersurfaces and half-lightlike submanifolds. Furthermore, screen distributions are also investigated, for any whose umbilical. Closely linked to above show, under some geometric conditions,...

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...

The properties of normal injectivity radius i(K, M) (thickness), of C submanifolds K of complete Riemannian manifolds M are studied. We introduce the notion of geometric focal distance for C submanifolds by using metric balls. A formula for i(K, M) in terms of the double critical points and the geometric focal distance is proved. The thickness of knots and ideal knots relate to the study of DNA...

An immersed submanifold f :M → R , n ≥ 3, into Euclidean space with the induced metric is called of rank two if at any point the kernel of its vector valued second fundamental form has codimension two. Equivalently, we have that the image of the Gauss map in the Grassmannian of non-oriented n-planes Gn is a surface. These submanifolds have been the object of a great deal of work in Riemannian G...