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arXiv math-ph/9909031math-phastro-phmath.DGmath.MPnucl-th
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黎曼椭球的规范理论

Gauge theory of Riemann ellipsoids

G. Rosensteel, J. Troupe

备注：
37 pages, 3 eps figures
发表信息：
J.Phys. A34 (2001) L169-L178
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摘要：

黎曼椭球的经典理论自然地表述为基于主G丛${\cal P}$的规范理论。结构群G=SO（3）是涡度群，丛${\cal P}=GL_+(3, R})$是一般线性群的连通分支。基流形是正定实$3\times 3$对称矩阵空间，在几何上与惯性椭球空间相同。丛${\cal P}$也是一个黎曼流形，其度规由动能决定。非完整约束决定了丛上的联系。特别是，平凡连接对应于刚体运动，自然黎曼连接-无旋流，和不变连接-下降的猫。曲率形式确定流体的场张量，这是一个类似的熟悉的法拉第张量。

英文摘要：

The classical theory of Riemann ellipsoids is formulated naturally as a gauge theory based on a principal G-bundle ${\cal P}$. The structure group G=SO(3) is the vorticity group, and the bundle ${\cal P}=GL_+(3, R})$ is the connected component of the general linear group. The base manifold is the space of positive-definite real $3\times 3$ symmetric matrices, identified geometrically with the space of inertia ellipsoids. The bundle ${\cal P}$ is also a Riemannian manifold whose metric is determined by the kinetic energy. Nonholonomic constraints determine connections on the bundle. In particular, the trivial connection corresponds to rigid body motion, the natural Riemannian connection - to irrotational flow, and the invariant connection - to the falling cat. The curvature form determines the fluid's field tensor which is an analogue of the familiar Faraday tensor.

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