We have yet to meet the star of the show. There is one object that we can place on a manifold whose importance dwarfs all others, at least when it comes to understanding gravity. This is the metric.
The existence of a metric brings a whole host of new concepts to the table which, collectively, are called Riemannian geometry. In fact, strictly speaking we will need a slightly different kind of metric for our study of gravity, one which, like the Minkowski metric, has some strange minus signs. This is referred to as Lorentzian Geometry and a slightly better name for this section would be “Introducing Riemannian and Lorentzian Geometry”. However, for our immediate purposes the differences are minor. The novelties of Lorentzian geometry will become more pronounced later in the course when we explore some of the physical consequences such as horizons.
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正文：英文 · OCR 机器稿 · 待校对
核对状态：OCR 机器稿 · 待校对
来源版本：michaelmas-2019-source-20260923
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