Reading Guide
One of the most comprehensive introductory surveys of the subject, charting the historical development from before Gauss to modern times, is by Berger (2004). Graduate-level treatments of the Riemannian connection can be found in Singer & Thorpe (1967), do Carmo (1976) and O'Neill (1997). Accessible introductions to Cartan's approach to connections using moving frames can be found in Ivey & Landsberg (2003) and Sharpe (1997). The classic treatment of connections can be found in Kobayashi & Nomizu (1963).
Read more about this topic: Riemannian Connection On A Surface
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