Abstract

Jacobian determines a bundle with total space consisting of orientation-preserving diffeomorphisms of a (connected) manifold over the space of positive functions on this manifold (with integral equal to volume for a compact manifold). It is proved that, for the \(n\)-sphere with standard metric, there is a unique connection on this bundle that is invariant with respect to all isometries of the sphere, and a description of this connection is given.

中文翻译：

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函数空间上作为束的微分同胚群的连接

摘要

雅可比行列式确定具有总空间的丛，该丛由（连接的）流形在该流形上的正函数空间上的方向保持微分同胚组成（对于紧凑流形，积分等于体积）。证明了对于具有标准度量的\(n\) -球体，在该丛上存在一个关于球体的所有等距不变的唯一连接，并给出了该连接的描述。