The natural operators lifting horizontal 1-forms to some vector bundle functors on fibered manifolds

There are completely described all \(\mathcal{VB}_{m,n}\)-gauge-natural operators \(C\) which, like to the Dorfman-Courant bracket, send closed linear \(3\)-forms \(H\in\Gamma^{l-\rm{clos}}_E(\bigwedge^3T^*E)\) on a smooth (\(\mathcal{C}^{\infty}\)) vector bundle \(E\) into \(\mathbf{R}\)-bilinear operators \[C_H:\Gamma^l_E(TE\oplus T^*E)\times \Gamma^l_E(TE\oplus T^*E)\to \Gamma^l_E(TE\oplus T^*E)\] transforming pairs of linear sections of \(TE\oplus T^*E\to E\) into linear sections of \(TE\oplus T^*E\to E\). Then all such \(C\) which also, like to the twisted Dorfman-Courant bracket, satisfy both some "restricted" condition and the Jacobi identity in Leibniz form are extracted.

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On the gauge-natural operators similar to the twisted Dorfman-Courant bracket

Opuscula Mathematica ◽

10.7494/opmath.2021.41.2.205 ◽

2021 ◽

Vol 41 (2) ◽

pp. 205-226

Author(s):

Włodzimierz M. Mikulski

Keyword(s):

Vector Bundle ◽

Jacobi Identity ◽

Courant Bracket ◽

Bilinear Operators ◽

Linear Sections ◽

Natural Operators

All \(\mathcal{VB}_{m,n}\)-gauge-natural operators \(C\) sending linear \(3\)-forms \(H \in \Gamma^{l}_E(\bigwedge^3T^*E)\) on a smooth (\(\mathcal{C}^\infty\)) vector bundle \(E\) into \(\mathbf{R}\)-bilinear operators \[C_H:\Gamma^l_E(TE \oplus T^*E)\times \Gamma^l_E(TE \oplus T^*E)\to \Gamma^l_E(TE \oplus T^*E)\] transforming pairs of linear sections of \(TE \oplus T^*E \to E\) into linear sections of \( TE \oplus T^*E \to E\) are completely described. The complete descriptions is given of all generalized twisted Dorfman-Courant brackets \(C\) (i.e. \(C\) as above such that \(C_0\) is the Dorfman-Courant bracket) satisfying the Jacobi identity for closed linear \(3\)-forms \(H\). An interesting natural characterization of the (usual) twisted Dorfman-Courant bracket is presented.

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Euler classes of vector bundles over manifolds

Mathematica Slovaca ◽

10.1515/ms-2017-0461 ◽

2021 ◽

Vol 71 (1) ◽

pp. 199-210

Author(s):

Aniruddha C. Naolekar

Keyword(s):

Vector Bundle ◽

Vector Bundles ◽

Euler Class ◽

Homology Sphere ◽

Rational Homology ◽

Rational Homology Sphere ◽

Orientable Manifolds

Abstract Let 𝓔 k denote the set of diffeomorphism classes of closed connected smooth k-manifolds X with the property that for any oriented vector bundle α over X, the Euler class e(α) = 0. We show that if X ∈ 𝓔2n+1 is orientable, then X is a rational homology sphere and π 1(X) is perfect. We also show that 𝓔8 = ∅ and derive additional cohomlogical restrictions on orientable manifolds in 𝓔 k .

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On spines of Seifert fibered manifolds

Aequationes Mathematicae ◽

10.1007/s000100300003 ◽

2003 ◽

Vol 65 (1) ◽

pp. 40-60 ◽

Cited By ~ 4

Author(s):

B. Ruini ◽

F. Spaggiari ◽

A. Vesnin

Keyword(s):

Fibered Manifolds

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WEAKLY UNIFORM RANK TWO VECTOR BUNDLES ON MULTIPROJECTIVE SPACES

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972711002243 ◽

2011 ◽

Vol 84 (2) ◽

pp. 255-260

Author(s):

EDOARDO BALLICO ◽

FRANCESCO MALASPINA

Keyword(s):

Vector Bundle ◽

Vector Bundles ◽

Splitting Type ◽

Multiprojective Spaces

AbstractHere we classify the weakly uniform rank two vector bundles on multiprojective spaces. Moreover, we show that every rank r>2 weakly uniform vector bundle with splitting type a1,1=⋯=ar,s=0 is trivial and every rank r>2 uniform vector bundle with splitting type a1>⋯>ar splits.

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