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Frame bundle
About: Frame bundle is a research topic. Over the lifetime, 1600 publications have been published within this topic receiving 23049 citations.
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TL;DR: In this article, a geometric construction of the Stiefel-Whitney classes when ξ is a real vector bundle and of the Chern classes when ǫ is a complex vector bundle is given.
4 citations
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TL;DR: In this paper, a special representation of an algebra bundle using Hochschild cohomology of an associative algebra bundle with coefficients in a bimodule bundle has been studied.
Abstract: Hochschild cohomology of an associative algebra bundle with coefficients in a bimodule bundle has been defined and studied in earlier paper. Here, by using cohomological methods, we establish that an algebra bundle is a semidirect product of its radical bundle and a semisimple subalgebra bundle. Further we define multiplication algebra bundle of an algebra bundle and representation of an algebra bundle. We study special representations of an algebra bundle using Hochschild cohomology of an associative algebra bundle with coefficients in a bimodule bundle. We observe that if a representation of an algebra bundle is special then its obstruction is zero. Further we show that a subgroup H of H2(ξ, N) is faithfully represented as a transitive group of translations operating on the set of those equivalence classes of algebra bundle extensions of ξ which determine a given representation [φ, K].
4 citations
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TL;DR: In this paper, the authors extend the Shatz stratification of sheaves to arbitrary families of projective schemes and show how the Harder-Narasimhan polygon of the restriction of the tangent bundle ΘIPn to space curves reflects the geometry of these curves and their embeddings.
Abstract: We extend the Shatz stratification of sheaves to arbitrary families of projective schemes. This allows a stratification of Hilbert schemes. We investigate how the Harder-Narasimhan polygon of the restriction of the tangent bundle ΘIPn to space curves reflects the geometry of these curves and their embeddings.
4 citations
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TL;DR: In this article, the authors define the notion of the $k$'th order differential (T^kg:T^kM\longrightarrow T^kN$) for a given differentiable map $g$ between manifolds $M$ and $N$.
Abstract: The tangent bundle $T^kM$ of order $k$, of a smooth Banach manifold $M$ consists of all equivalent classes of curves that agree up to their accelerations of order $k$. In the previous work of the author he proved that $T^kM$, $1\leq k\leq \infty$, admits a vector bundle structure on $M$ if and only if $M $ is endowed with a linear connection or equivalently a connection map on $T^kM$ is defined. This bundle structure depends heavily on the choice of the connection. In this paper we ask about the extent to which this vector bundle structure remains isomorphic. To this end we define the notion of the $k$'th order differential $T^kg:T^kM\longrightarrow T^kN$ for a given differentiable map $g$ between manifolds $M$ and $N$. As we shall see, $T^kg$ becomes a vector bundle morphism if the base manifolds are endowed with $g$-related connections. In particular, replacing a connection with a $g$-related one, where $g:M\longrightarrow M$ is a diffeomorphism, follows invariant vector bundle structures. Finally, using immersions on Hilbert manifolds, convex combination of connection maps and manifold of $C^r$ maps we offer three examples to support our theory and reveal its interaction with the known problems such as Sasaki lift of metrics.
4 citations