Trivial Fiber Bundle Research Articles

The group-theoretical classification of the 11-dimensional classical homogeneous Kaluza–Klein cosmologies

In the context of the classical Kaluza–Klein cosmology the generalized Bianchi models in 11 dimensions are considered. These are space-times whose spacelike ten-dimensional sections are the hypersurfaces of transitivity for a ten-dimensional isometry group of the total space-time. Such a space-time is a trivial principal fiber bundle P(M,G7), where M is a four-dimensional physical space-time with an isometry group G3 (of a Bianchi type) and G7 is a compact isometry group of the compact internal space. The isometry group of P is G10=G3⊗G7, hence all the generalized Bianchi models are classified by enumerating the relevant groups G7. Due to the compactness of G7 the result is astonishingly simple: there are three distinct homogeneous internal spaces in addition to the 11 ordinary Bianchi types for M.

The canonical foliations of a locally conformal Kähler manifold

Let M be a locally conformal Kahler manifold. Then the Kahler form Ω of M satisfies dΩ=ωΩ for some closed 1 -form ω, called the Lee form of M. We show that M admits three canonical foliations (four if ω is parallel) and we prove several properties of them, improving previous results of I. Vaisman. In particular all of these foliations are totally geodesic and Riemannian, and one of them is also almost complex. If this latter foliation is regular on a compact M, then we prove that M is a locally trivial fiber bundle over a compact Kahler manifold M, and the fibers are totally geodesic flat 2-tori. Finally we study geometrical properties, the canonical class and the Godbillon-Vey class of the totally real foliation of a CR-submanifold N ⊂cM.

General prolongations and (x, t)-depending pseudopotentials for the KdV equation

Given an exterior differential system on a manifold M, we study general prolongations of the system on a locally trivial fiber bundle (M̃, π̃, M) by a Cartan–Ehresmann connection. We characterize such prolongations for the system associated with the KdV equation without any assumption of ‘‘(x, t) independence.’’ The partial Lie algebra discovered by Wahlquist–Estabrook [J. Math. Phys. 16, 1 (1975)] appears by this way as an intrinsic tool. Simple analytic pseudopotentials are classified up to diffeomorphism.

Stability for infinite-dimensional fibre bundles

In this paper, we prove that any locally trivial fibre bundles $p:X \to B$ with fibre $M$ a manifold modeled on an infinite-dimensional space $E$ (e.g. the Hilbert space $l_2$ or the Hilbert cube $Q$) is bundle isomorphic to the bundle $p * \mathrm{proj}:X \times E \to B$. Further, we can obtain a strong version of this Bundle Stability Theorem. From Bundle Stability Theorem, we can introduce the notion of deficiency in bundles. We show that a finite union of locally deficient sets is deficient and we prove a bundle version of Mapping Replacement Theorem.

Isometrische Immersionen der Kodimension 2 von Raumformen

Let M, resp.\(\tilde M\), denote Riemannian manifolds of dimensions m>4, resp.\(\tilde m\)=m+2, and of constant sectional curvatures C, resp.\(\tilde C\), with\(\tilde C\) 0 and\(\tilde M\) is a standard space form, then the foliation L is a (globally) trivial fibre bundle with fibre Sm−1.

Locally trivial homotopy functors and their domain of representation

AbstractThe well known theorem of E. H. Brown on the representation of Homotopy Functors is extended to a larger domain category. This is accomplished by requiring the functor to satisfy a further condition, that of being locally trivial: the condition is both necessary and sufficient for the extension and is satisfied by functors arising from locally trivial fibre bundles such as K-theory.

K (π n )-Torsors and the Interpretation of “Triple” Cohomology

A categorical, semi-simplicial interpretation theory for "triple" cohomology is outlined which uses the groups TORS(U) (n)[X;pi] of connected components of certain categories of K(pi,n)-torsors (higher dimensional generalizations of locally trivial principal fiber bundles) to interpret the cohomology groups, i.e., TORS(U) (n)[X;pi] [unk] H(n)(X;pi), n >/= 1.

Foliations transverse to fibers of a bundle

Consider a fiber bundle where the base space and total space are compact, connected, oriented smooth manifolds and the projection map is smooth. It is shown that if the fiber is null-homologous in the total space, then the existence of a foliation of the total space which is transverse to each fiber and such that each leaf has the same dimension as the base implies that the fundamental group of the base space has exponential growth. Introduction. Let (E, p, B) be a locally trivial fiber bundle where /»:£-*-/? is the projection, E and B are compact, connected, oriented smooth manifolds and p is a smooth map. (By smooth we mean C for some r^l and, henceforth, all maps are assumed smooth.) Let b denote the dimension of B and k denote the dimension of the fiber (thus, dim E= b+k). By a section of the fibration we mean a smooth map a:B-+E such that p o a=idB. It is well known that if a section exists then the fiber over any point in B represents a nontrivial element in Hk(E; Z) since the image of a section is a compact orientable manifold which has intersection number one with the fiber over any point in B. The notion of a section may be generalized as follows. Definition. A polysection of (E,p,B) is a covering projection tr:B-*B together with a map C:B-*E such that the following diagram commutes. Note that if the covering projection is a diffeomorphism then the map f o 7T_1 is a section in the usual sense. One important situation in which polysections arise is the following. Suppose that F is a smooth foliation of E such that each leaf of F is Received by the editors April 4, 1973 and, in revised form, May 1, 1973. AMS (MOS) subject classifications (1970). Primary 57D30; Secondary 55F10.

The Signature of Fiber Bundles

Let $Y\hookrightarrow Z\underrightarrow \pi X$ be a locally trivial fiber bundle in the category of oriented topological manifolds. It is shown that if the identity component of the structure group $G$ has finite index, then (signature of $Z$)=(signature of $X$) $\blacksquare$ (signature of $Y$).

The signature of fiber bundles

Let Y ↪ Z π → X Y\hookrightarrow Z\underrightarrow \pi X be a locally trivial fiber bundle in the category of oriented topological manifolds. It is shown that if the identity component of the structure group G G has finite index, then (signature of Z Z )=(signature of X X ) ◼ \blacksquare (signature of Y Y ).