A Formula for the Tangent Bundle of Flag Manifolds and Related Manifolds

Lam, Kee Yuen

doi:10.2307/1998048

Cited by 17 publications

(22 citation statements)

References 9 publications

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“…Theorem 2.8. (W. Sutherland [83], K. Y. Lam [53], D. Handel [28].) The real, complex, and quaternionic Stiefel manifolds V n,k , W n,k , Z n,k are parallelizable when k ≥ 2.…”

Section: Singhof and Wemmermentioning

confidence: 99%

“…Although one has the notion of quaternionic projective Stiefel manifolds, not much is known about their span. (See [53].) For this reason we shall be contend with defining them, but discuss the vector field problem only for real and complex projective Stiefel manifolds.…”

Section: Singhof and Wemmermentioning

confidence: 99%

“…The following descriptions of the tangent bundle of real and complex projective Stiefel manifolds was obtained by Lam [53].…”

Section: Singhof and Wemmermentioning

confidence: 99%

“…as Z(F)-vector bundles where Z(F) denotes the centre of the division ring F. We refer the reader to [53] for a proof.…”

Section: 5mentioning

confidence: 99%

“…. , 1) was first proved by Lam [53], making essential use of the functor µ 2 , which is an analogue of the second exterior power in the real and complex case.…”

Section: 5mentioning

confidence: 99%

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The Vector Field Problem for Homogeneous Spaces

Sankaran

2019

Trends in Mathematics

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Dedicated to Professor Peter Zvengrowski with admiration and respect.Abstract. Let M be a smooth connected manifold of dimension n ≥ 1. A vector field on M is an association p → v(p) of a tangent vector v(p) ∈ T p M for each p ∈ M which varies continuously with p. In more technical language it is a (continuous) cross-section of the tangent bundle τ (M ). The vector field problem asks: Given M , what is the largest possible number r such that there exist vector fields v 1 , . . . , v r which are everywhere linearly independent, that is, v 1 (x), . . . , v r (x) ∈ T x M are linearly independent for every x ∈ M . The number r is called the span of M , written span(M ). It is clear that 0 ≤ span(M ) ≤ dim(M ). The vector field problem is an important and classical problem in differential topology. In this survey we shall consider the vector field problem focussing mainly on the class of compact homogeneous spaces.2010 Mathematics Subject Classification. 57R25.

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“…Theorem 2.8. (W. Sutherland [83], K. Y. Lam [53], D. Handel [28].) The real, complex, and quaternionic Stiefel manifolds V n,k , W n,k , Z n,k are parallelizable when k ≥ 2.…”

Section: Singhof and Wemmermentioning

confidence: 99%

Section: Singhof and Wemmermentioning

confidence: 99%

“…The following descriptions of the tangent bundle of real and complex projective Stiefel manifolds was obtained by Lam [53].…”

Section: Singhof and Wemmermentioning

confidence: 99%

“…as Z(F)-vector bundles where Z(F) denotes the centre of the division ring F. We refer the reader to [53] for a proof.…”

Section: 5mentioning

confidence: 99%

“…. , 1) was first proved by Lam [53], making essential use of the functor µ 2 , which is an analogue of the second exterior power in the real and complex case.…”

Section: 5mentioning

confidence: 99%

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The Vector Field Problem for Homogeneous Spaces

Sankaran

2019

Trends in Mathematics

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Equivariant cobordism of Grassmann and flag manifolds

Mukherjee¹

1995

Proc. Indian Acad. Sci. (Math. Sci.)

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Degrees of maps between locally symmetric spaces

Mondal

Sankaran

2016

Bulletin des Sciences Mathématiques

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Abstract. Let X be a locally symmetric space Γ\G/K where G is a connected noncompact semisimple real Lie group with trivial centre, K is a maximal compact subgroup of G, and Γ ⊂ G is a torsion-free irreducible lattice in G. Let Y = Λ\H/L be another such space having the same dimension as X. Suppose that real rank of G is at least 2. We show that any f : X → Y is either null-homotopic or is homotopic to a covering projection of degree an integer that depends only on Γ and Λ. As a corollary we obtain that the set [X, Y ] of homotopy classes of maps from X to Y is finite.We obtain results on the (non-) existence of orientation reversing diffeomorphisms on X as well as the fixed point property for X.

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A Formula for the Tangent Bundle of Flag Manifolds and Related Manifolds

Cited by 17 publications

References 9 publications

The Vector Field Problem for Homogeneous Spaces

The Vector Field Problem for Homogeneous Spaces

Equivariant cobordism of Grassmann and flag manifolds

Degrees of maps between locally symmetric spaces

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