Geometry of the Total Space of a Vector Bundle

1994 ◽  
pp. 35-65
Author(s):  
Radu Miron ◽  
Mihai Anastasiei
Keyword(s):  
Vector Bundle ◽  
Total Space

Vector bundles that fill n-space

1965 ◽  
Vol 61 (4) ◽  
pp. 869-875 ◽  
Author(s):  
S. A. Robertson ◽  
R. L. E. Schwarzenberger
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Total Space

The idea of exact filling bundle may be described roughly as follows. Suppose that ξk is a vector bundle with fibre Rk, total space E(ξk) and base X. We say that ξk is a real k-plane bundle on X. Let in be the trivial n-plane bundle on X so that E(in) = X × Rn. A bundle monomorphism j: ξk → in defines a map : E(ξk)→Rn obtained by composition of the embedding E(ξk)→E(in) and the product projection E(in) → Rn. The map represents each fibre of ξk as a k-plane in Rn.

The hypoelliptic Laplacian on X = G/K

2011 ◽  
Author(s):  
Jean-Michel Bismut
Keyword(s):  
Vector Bundle ◽  
Symmetric Space ◽  
Dirac Operator ◽  
Clifford Algebras ◽  
Second Order ◽  
Total Space ◽  
Hypoelliptic Operator ◽  
Heisenberg Algebras

This chapter constructs the hypoelliptic Laplacian ℒbX > 0 acting on the total space of a vector bundle TX ⊕ N ≃ g over the symmetric space X = G/K. The operator ℒbX is obtained using general constructions involving Clifford algebras and Heisenberg algebras, and also the Dirac operator of Kostant. The end result is the elliptic Laplacian 𝓛 X on X as well as the hypoelliptic Laplacian ℒbX, which is a second order hypoelliptic operator acting on X^. Among other things, this chapter gives a key formula relating 𝓛 XℒbX, as well as various formulas involving the operator ℒbX+La.

scholarly journals Structure presque-transverse intégrable

1988 ◽  
Vol 37 (2) ◽  
pp. 173-177
Author(s):  
Tong Van Duc
Keyword(s):  
Vector Bundle ◽  
Total Space ◽  
Transversal Structure

We find conditions for which a manifold endowed with an integrable almost transversal structure is the total space of a vector bundle isomorphic to the transversal bundle of a foliation.

scholarly journals Variation formulas for principal functions, II: Applications to variation for harmonic spans

2011 ◽  
Vol 204 ◽  
pp. 19-56 ◽  
Author(s):  
Sachiko Hamano ◽  
Fumio Maitani ◽  
Hiroshi Yamaguchi
Keyword(s):  
Direct Application ◽  
Total Space ◽  
Geometric Meaning ◽  
Principal Function ◽  
Variation Formula ◽  
Circular Slit ◽  
Moving Domain

AbstractA domainD⊂ Czadmits the circular slit mappingP(z) fora, b∈Dsuch thatP(z) – 1/(z–a) is regular ataandP(b) = 0. We callp(z) =log|P(z)|theLi-principal functionandα= log |P′(b)| theL1-constant, and similarly, the radial slit mappingQ(z) implies theL0-principal functionq(z) and theL0-constantβ. We calls=α–βtheharmonic spanfor (D, a, b). We show the geometric meaning ofs. Hamano showed the variation formula for theL1-constantα(t) for the moving domainD(t) in Czwitht∈B:= {t∈ C: |t| <ρ}. We show the corresponding formula for theL0-constantβ(t) forD(t) and combine these to prove that, if the total spaceD =∪t∈B(t, D(t)) is pseudoconvex inB× Cz, thens(t) is subharmonic onB. As a direct application, we have the subharmonicity of log coshd(t) onB, whered(t) is the Poincaré distance betweenaandbonD(t).

Euler classes of vector bundles over manifolds

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 .

scholarly journals WEAKLY UNIFORM RANK TWO VECTOR BUNDLES ON MULTIPROJECTIVE SPACES

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.

scholarly journals An inverse problem for a hyperbolic system on a vector bundle and energy measurements

2011 ◽  
Vol 354 (4) ◽  
pp. 1431-1464
Author(s):  
Katsiaryna Krupchyk ◽  
Matti Lassas
Keyword(s):  
Inverse Problem ◽  
Vector Bundle ◽  
Hyperbolic System
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