Exceptional simple lie groups and related topics in recent differential geometry

Murakami, Shingo

doi:10.1007/bfb0087534

Cited by 23 publications

(21 citation statements)

References 7 publications

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“…By the principle of triality in SO.8; C/ (see [9,11,17] where I 8 stands for the unit matrix of SO.8; C/. The restriction of ' to Spin.8/ becomes the universal covering map of SO.8/.…”

Section: Preliminariesmentioning

confidence: 99%

Visible actions on the non-symmetric homogeneous space SO(8, ℂ)/G 2(ℂ)

Sasaki¹

2011

Advances in Pure and Applied Mathematics

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“…By the principle of triality in SO.8; C/ (see [9,11,17] where I 8 stands for the unit matrix of SO.8; C/. The restriction of ' to Spin.8/ becomes the universal covering map of SO.8/.…”

Section: Preliminariesmentioning

confidence: 99%

Visible actions on the non-symmetric homogeneous space SO(8, ℂ)/G 2(ℂ)

Sasaki¹

2011

Advances in Pure and Applied Mathematics

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“…Much of the background can be found in [15]; we use the conventions found in [11]. The octonions O are a non-associative normed division algebra of dimension 8 over R. The octionions are alternative, meaning that the subalgebra generated by any two elements of O is associative.…”

Section: G and The Octonionsmentioning

confidence: 99%

Almost Positive Curvature on an Irreducible Compact Rank 2 Symmetric Space

DeVito

Nance

2018

International Mathematics Research Notices

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A Riemannian manifold is said to be almost positively curved if the sets of points for which all 2-planes have positive sectional curvature is open and dense. We show that the Grassmannian of oriented 2planes in R 7 admits a metric of almost positive curvature, giving the first example of an almost positively curved metric on an irreducible compact symmetric space of rank greater than 1. The construction and verification rely on the Lie group G 2 and the octonions, so do not obviously generalize to any other Grassmannians. arXiv:1707.07590v1 [math.DG]

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“…But then the triple (A q , B q , C q ) belongs to a Spin(7) subgroup of Spin(8) and hence any of A q , B q , C q has to belong to a SO(7) ⊂ SO(8) (cf. [Mur89,page 194]). Recall on the other hand that any subgroup SO(7) ⊂ SO(8), when acting on the sphere S 7 , admits a fixed point and it is conjugate with the standard SO(7) (cf.…”

Section: Examplesmentioning

confidence: 99%

Spin(9) geometry of the octonionic Hopf fibration

Parton

Piccinni

Vuletescu

2013

Transformation Groups

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We deal with Riemannian properties of the octonionic Hopf fibration S 15 → S 8 , in terms of the structure given by its symmetry group Spin(9). In particular, we show that any vertical vector field has at least one zero, thus reproving the non-existence of S 1 subfibrations. We then discuss Spin(9)-structures from a conformal viewpoint and determine the structure of compact locally conformally parallel Spin(9)-manifolds. Eventually, we give a list of examples of locally conformally parallel Spin(9)-manifolds.2010 Mathematics Subject Classification. Primary 53C26, 53C27, 57R25.

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Exceptional simple lie groups and related topics in recent differential geometry

Cited by 23 publications

References 7 publications

Visible actions on the non-symmetric homogeneous space SO(8, ℂ)/G 2(ℂ)

Visible actions on the non-symmetric homogeneous space SO(8, ℂ)/G 2(ℂ)

Almost Positive Curvature on an Irreducible Compact Rank 2 Symmetric Space

Spin(9) geometry of the octonionic Hopf fibration

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