标题： 外部广义几何 显示英文标题

标题： Exterior Generalised Geometry

摘要： 本文的目的是将用于研究半黎曼浸入的广义几何工具进行推广，并在某些情况下专门针对半黎曼超曲面。 给定一个精确的Courant代数束$E \to M$和一个浸入$\iota\colon N \hookrightarrow M$，存在一个已知的构造，即精确的Courant代数束$\iota^! E \to N$，它是$E$的拉回。 本文解释了广义度量和散度算子的拉回。 假设$N$是一个超曲面，它发展了广义外曲率的概念，引入了广义第二基本形式和广义平均曲率。 得到了广义Gau{\ss }-Codazzi方程。 作为应用，在广义几何的形式中建立了广义爱因斯坦方程初始值公式的约束方程。 进一步的应用包括超曲面的基本定理的广义几何版本以及广义Kähler和超Kähler结构限制在与广义几乎复结构相容的子流形上的结果。 特别是，我们表征了相对于规范广义联络为平坦的精确半黎曼Courant代数束。 这些在上述基本定理中扮演环境空间的角色。 显示英文摘要

摘要： It is the aim of this paper to transfer to generalised geometry tools employed in the study of semi-Riemannian immersions, specializing at times to semi-Riemannian hypersurfaces. Given an exact Courant algebroid $E \to M$ and an immersion $\iota\colon N \hookrightarrow M$, there is a well-known construction of an exact Courant algebroid $\iota^! E \to N$, the pullback of $E$. This paper explains the pullback of generalised metrics and divergence operators. Assuming $N$ is a hypersurface, it develops the notion of generalised exterior curvature, introducing the generalised second fundamental form and the generalised mean curvature. Generalised versions of the Gau{\ss}-Codazzi equations are obtained. As an application, the constraint equations for the initial value formulation of the generalised Einstein equations are established in the formalism of generalised geometry. Further applications include a generalised geometry version of the fundamental theorem for hypersurfaces and the result that generalised K\"ahler and hyper-K\"ahler structures restrict to submanifolds compatible with the generalised almost complex structure. In particular, we characterise exact semi-Riemannian Courant algebroids which are flat with respect to the canonical generalised connection. These play the role of the ambient space in the fundamental theorem mentioned above.