Connections and curvature in the Riemannian geometry of configuration spaces (Q5952323)
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scientific article; zbMATH DE number 1688728
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Connections and curvature in the Riemannian geometry of configuration spaces |
scientific article; zbMATH DE number 1688728 |
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Connections and curvature in the Riemannian geometry of configuration spaces (English)
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4 March 2003
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configuration spaces
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Poisson spaces
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covariant derivatives
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curvature
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Let \(M\) be a compact Riemannian manifold. Consider the configuration space \(\Gamma\) over \(M\), that is NEWLINE\[NEWLINE\Gamma= \Biggl\{\gamma= \sum_i \delta_{x_i};\;\{x_i\}\text{ are different points in }M\Biggr\},NEWLINE\]NEWLINE where \(\delta_x\) denotes the Dirac mass at the point \(x\in M\).NEWLINENEWLINEThe analysis over \(\Gamma\) has been developed [see for example, \textit{M. Röckner}, Stud. Adv. Math. 8, 157--231 (1998; Zbl 1037.58026)]. The main purpose of this work is to develop the differential geometry over \(\Gamma\) in the spirit of \textit{A.-B. Cruzeiro} and \textit{P. Malliavin} [see J. Funct. Anal. 139, 119--181 (1996; Zbl 0869.60060)].
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