Almost-Complex Substructures on the Sphere

Dibag, I.

doi:10.2307/2041342

Cited by 4 publications

(2 citation statements)

References 3 publications

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“…In the papers [3], [5], [12], [18], [19] and [23] the problem of when the structure group G n in one of the fibrations (1.1) -(1.3) can be reduced to a group G = SU(k) or Sp(k) via a standard inclusion G ֒→ G n was solved and the results were expressed in terms of complex and quaternionic James numbers in a way similar to the result quoted above. The following theorem can be regarded as a generalization of these results.…”

Section: Resultsmentioning

confidence: 99%

“…Considering standard inclusions ρ : G = G k ֒→ G n we get the famous problem on sections of Stiefel manifolds over spheres resolved in [1], [3], [5] and [23]. The other standard inclusions SU(k) ֒→ SO(n), Sp(k) ֒→ SO(n) and Sp(k) ֒→ SU(n) are dealt with in [12], [18] and [19], respectively. In these cases the question was to find a minimal standard subgroup to which G n can be reduced.…”

Section: Introductionmentioning

confidence: 99%

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$G$-Structures on Spheres

Čadek

Crabb

2006

Proc. Lond. Math. Soc.

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This paper deals with the problem of determining those groups G and the homomorphisms ρ to which the structure group G n in the fibrations above can be reduced. The problem has been solved in many interesting special cases. Considering standard inclusions ρ : G = G k → G n we get the famous problem on sections of Stiefel manifolds over spheres resolved in [1, 3, 5, 23]. The other standard inclusions SU(k) → SO(n), Sp(k) → SO(n) and Sp(k) → SU(n) are dealt with in [12], [18] and [19], respectively. In these cases the question was to find a minimal standard subgroup to which G n can be reduced.In [15] Leonard asked an opposite question: find all maximal proper subgroups to which G n can be reduced. He solved it in the cases when G is a reducible maximal subgroup of G n . Moreover, he proved that G n cannot be reduced to any proper subgroup (G, ρ) if (1) n is even and G n = SO(n) or SU(n), unless G n = SO(6) and G = SU(3);(2) n ≡ 11 mod 12 and G n = Sp(n);(3) G is a non-simple irreducible maximal proper subgroup of G n . 793 can be reduced to G via a homomorphism ρ : G → G n if and only if one of the following cases occurs: (A) G n = SO(n), G = SO(k), n = m − 1, m ≡ 0 mod a(m − k) and, up to conjugation, ρ is the standard inclusion SO(k) → SO(n); (B) G n = SO(n), G = SU(k), n = 2m − 1, m ≡ 0 mod 2 ν2(b(m−k)) and, up to conjugation, ρ is the composition of the standard inclusions SU(k) → SO(2k) → SO(n) or the composition SU(4) → SO(8) × SO(6) → SO(15) where the first homomorphism is given on the first factor by the standard inclusion and on the second factor by the double covering SU(4) ∼ = Spin(6) → SO(6); (C) G n = SO(n), G = Sp(k), n = 4m − 1, m ≡ 0 mod 2 ν2(c(m−k)) and, up to conjugation, ρ is the composition of the standard inclusions Sp(k) → SO(4k) → SO(n) or the exterior square Sp(3) → SO(15); (D) G n = SU(n), G = SU(k), n = m − 1, m ≡ 0 mod b(m − k) and, up to conjugation, ρ is the standard inclusion SU(k) → SU(n); (E) G n = SU(n), G = Sp(k), n = 2m − 1, m ≡ 0 mod c(m − k) and, up to conjugation, ρ is the composition of the standard inclusions Sp(k) → SU(2k) → SU(n); (F) G n = Sp(n), G = Sp(k), n = m − 1, m ≡ 0 mod c(m − k) and, up to conjugation, ρ is the standard inclusion Sp(k) → Sp(n).As a consequence of Theorem 2.

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

$G$-Structures on Spheres

Čadek

Crabb

2006

Proc. Lond. Math. Soc.

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On quaternionic James numbers and almost-quaternion substructures on the sphere

Önder¹

1982

Proc. Amer. Math. Soc.

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Almost-quaternion (𝑚-1)-substructures on 𝑆^{4𝑚-3}

Önder¹

1984

Proc. Amer. Math. Soc.

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Abstract. We prove that S4"'~3 admits an almost-quaternion (m -l)-substructure if and only if m = 2, completing the missing case in our paper On quaternionic James numbers and almost-quaternion substructures on the sphere.In [2], an almost-quaternion substructure on an orientable zz-manifold M is defined as the reduction of the structure group of T(M) from SO(n) to Sp(k) X SO(n -4k). This is equivalent to the existence of a 4zc-dimensional subbundle of T(M) together with two normalized almost complex structure maps F, G (see [1]), defined on the total space of the subbundle such that FG = -GF. In [2] we have given a theorem which describes the values of zc and n such that S" admits an almost-quaternion ^-substructure. There, all the cases were covered except for the case n = 4zrz -3 and k = m -\ for some zrz. In this note we prove the following theorem about this case:' Theorem. s4m~3 admits an almost-quaternion (m -\)-substructure if and only if m = 2.

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Almost-Complex Substructures on the Sphere

Abstract: Abstract.The paper solves completely the existence problem of almostcomplex substructures on spheres.

Cited by 4 publications

References 3 publications

$G$-Structures on Spheres

$G$-Structures on Spheres

On quaternionic James numbers and almost-quaternion substructures on the sphere

Almost-quaternion (𝑚-1)-substructures on 𝑆^{4𝑚-3}

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