On noncommutative and pseudo-Riemannian geometry
- 作者：Alexander Strohmaier
- 关键词：Pseudo-Riemannian ; Noncommutative tori ; Noncommutative geometry
- 刊名：Journal of Geometry and Physics
- 出版年：2006
- 期刊代码：29_03930440
- 类别：ph
- 出版时间：2006
- 卷：56
- 期：2
- 页码：175-195
- 文件大小：262 K
摘要
We introduce the notion of a pseudo-Riemannian spectral triple which generalizes the notion of spectral triple and allows for a treatment of pseudo-Riemannian manifolds within a noncommutative setting. It turns out that the relevant spaces in noncommutative pseudo-Riemannian geometry are not Hilbert spaces any more but Krein spaces, and Dirac operators are Krein-selfadjoint. We show that the noncommutative tori can be endowed with a pseudo-Riemannian structure in this way. For the noncommutative tori as well as for pseudo-Riemannian spin manifolds the dimension, the signature of the metric, and the integral of a function can be recovered from the spectral data.