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Linear connection

2 bytes added, 17:29, 6 January 2012
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===Definition part (pointwise form)===
A '''linear connection''' is a smooth choice <math>\nabla</math> of the following: at each point <math>p \in M</math>, there is a map <math>{}^p\nabla: T_p(M) \times \Gamma(TM) \to T_p(M)</math>, satisfying some conditions. The map is written as <math>{}^p\nabla_X(v)</math> where <math>X \in T_p(M)</math> and <math>v \in \Gamma(ETM)</math>.
* It is <math>\R</math>-linear in <math>X</math> (that is, in the <math>T_p(M)</math> coordinate).
* It is <math>\R</math>-linear in <math>\Gamma(TM)</math> (viz the space of sections on <math>ETM</math>).
* It satisfies the following relation called the Leibniz rule:
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