G_2流形的形变
Deformations in G_2 Manifolds
摘要：
本文研究了G_2流形M^7中具有校准3-形式流形的结合子流形的变形。选择M上的2-平面场Λ（它总是存在）将M的切丛分裂为一个3维缔合丛和一个复4-平面丛TM= E \oplus V的直和，这有助于我们将形变与Seiberg-Witten型方程联系起来。本文仅用向量积运算（等价积）证明了关于G_2流形的已有结果和新结果。我们认为，把丰富的G_2几何的各种不同的局部标识（如向量积、表示论和八元数代数）混合起来，使G_2流形的研究看起来比实际困难（如McLean定理\cite{m}的证明）。我们相信这里的方法可以让事情变得更容易，并保持演示的基本性。本文件基本上是自成一体的。
英文摘要：
Here we study the deformations of associative submanifolds inside a G_2 manifold M^7 with a calibration 3-form ϕ. A choice of 2-plane field Λon M (which always exits) splits the tangent bundle of M as a direct sum of a 3-dimensional associate bundle and a complex 4-plane bundle TM= E\oplus V, and this helps us to relate the deformations to Seiberg-Witten type equations. Here all the surveyed results as well as the new ones about G_2 manifolds are proved by using only the cross product operation (equivalently ϕ). We feel that mixing various different local identifications of the rich G_2 geometry (e.g. cross product, representation theory and the algebra of octonions) makes the study of G_2 manifolds looks harder then it is (e.g. the proof of McLean's theorem \cite{m}). We believe the approach here makes things easier and keeps the presentation elementary. This paper is essentially self contained.
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