In this paper we give a survey of various results about the topology of oriented Grassmannian bundles related to the exceptional Lie group G 2 . Some of these results are new. We give self-contained proofs here. One often encounters these spaces when studying submanifolds of manifolds with calibrated geometries. For the sake of completeness we decided to collect them here in a self-contained way to be easily accessible for future usage in calibrated geometry. As an application we deduce existence of certain special 3 and 4 dimensional submanifolds of G 2 manifolds with special properties, which appear in the first named author's work with S. Salur about G 2 dualities.
IntroductionRecall that G 2 ⊂ SO (7) is the 14-dimensional exceptional Lie group defined as the automorphisms of the imaginary octonions im(O) = R 7 preserving the cross product operation R 7 × R 7 → R 7 (e.g. [HL], [Br], [AS1], [AS2]). Octonions are the elements of the 8 dimensional division algebra O = H ⊕ lH = R 8 where H are the quaternions, O is generated by 1, i, j, k, l, li, lj, lk . The cross product operation × on im(O) is induced from the octonion multiplication on O by uWe say an oriented 7-manifold M 7 has a G 2 structure if its SO(7)-tangent frame bundle lifts to a G 2 -bundle by the canonical fibration:Alternatively G 2 can be defined by the special 3-frames in R 7 as follows:or as linear automorphisms of R 7 preserving a certain 3-form ϕ 0 ∈ Ω 7 (R 7 )By using this last definition, a G 2 structure on M 7 can be defined as a 3form ϕ ∈ Ω 3 (M 7 ) such that at each p ∈ M the pair (T p (M), ϕ(p)) is (pointwise)
In this paper, we study some classes of submanifolds of codimension one and two in the Page space. These submanifolds are totally geodesic. We also compute their curvature and show that some of them are constant curvature spaces. Finally we give information on how the Page space is related to some other metrics on the same underlying smooth manifold.Orta mh. Z übeyde Hanim cd. No 5-3 Merkez 74100 Bartin, T ürkíye.
In this article, we give a survey of Geometric Invariant Theory for Toric Varieties, and present an application to the Einstein-Weyl Geometry. We compute the image of the Minitwistor space of the Honda metrics as a categorical quotient according to the most efficient linearization. The result is the complex weighted projective space CP 1,1,2 . We also find and classify all possible quotients.
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