23,111 research outputs found
Holonomy and submanifold geometry
We survey applications of holonomic methods to the study of submanifold geometry, showing the consequences of some sort of extrinsic version of de Rham decomposition and Berger's Theorem, the so-called Normal Holonomy Theorem. At the same time, from geometric methods in submanifold theory we sketch very strong applications to the holonomy of Lorentzian manifolds. Moreover we give a conceptual modern proof of a result of Kostant for homogeneous space
Surfaces, submanifolds, and aligned Fox reimbedding in non-Haken 3-manifolds
Understanding non-Haken 3-manifolds is central to many current endeavors in
3-manifold topology. We describe some results for closed orientable surfaces in
non-Haken manifolds, and extend Fox's theorem for submanifolds of the 3-sphere
to submanifolds of general non-Haken manifolds. In the case where the
submanifold has connected boundary, we show also that the boundary-connected
sum decomposition of the submanifold can be aligned with such a structure on
the submanifold's complement.Comment: 10 page
Reducibility of complex submanifolds of the complex euclidean spaces
Let M be a simply connected complex submanifold of CN. We prove that M is irreducible, up a totally geodesic factor,if and only if the normal holonomy group acts irreducibly. This is an extrinsic analogue of the well-known De Rham decomposition theorem for a complex manifold. Our result is not valid in the real context, as it is shown by many counterexamples
Hyperbolic dynamics of discrete dynamical systems on pseudo-riemannian manifolds
We consider a discrete dynamical system on a pseudo-Riemannian manifold and
we determine the concept of a hyperbolic set for it. We insert a condition in
the definition of a hyperbolic set which implies to the unique decomposition of
a part of tangent space (at each point of this set) to two unstable and stable
subspaces with exponentially increasing and exponentially decreasing dynamics
on them. We prove the continuity of this decomposition via the metric created
by a torsion-free pseudo-Riemannian connection. We present a global attractor
for a diffeomorphism on an open submanifold of the hyperbolic space
which is not a hyperbolic set for it
Warped products and Spaces of Constant Curvature
We will obtain the warped product decompositions of spaces of constant
curvature (with arbitrary signature) in their natural models as subsets of
pseudo-Euclidean space. This generalizes the corresponding result by S. Nolker
to arbitrary signatures, and has a similar level of detail. Although our
derivation is complete in some sense, none is proven. Motivated by
applications, we will give more information for the spaces with Euclidean and
Lorentzian signatures. This is an expository article which is intended to be
used as a reference. So we also give a review of the theory of circles and
spheres in pseudo-Riemannian manifolds
Duality in a special class of submanifolds and Frobenius manifolds
We prove a duality principle for a special class of submanifolds in
pseudo-Euclidean spaces. This class of submanifolds with potential of normals
is introduced in this paper. We prove also, for example, that an arbitrary
Frobenius manifold can be realized as a certain flat submanifold of this very
natural class.Comment: 3 page
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