Linear connection

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This lives as an element of: the space of \R-bilinear maps \Gamma(TM) \times \Gamma(TM) \to \Gamma(TM)

Definition

Given data

Definition part (pointwise form)

A linear connection is a smooth choice \nabla of the following: at each point p \in M, there is a map {}^p\nabla: T_p(M) \times \Gamma(TM) \to T_p(M), satisfying some conditions. The map is written as {}^p\nabla_X(v) where X \in T_p(M) and v \in \Gamma(TM).

  • It is \R-linear in X (that is, in the T_p(M) coordinate).
  • It is \R-linear in \Gamma(TM) (viz the space of sections on TM).
  • It satisfies the following relation called the Leibniz rule:

{}^p\nabla_X(fv) = (Xf)(p) (v) + f(p) ^p\nabla_X(v)

Definition part (global form)

A linear connection is a map \nabla:\Gamma(TM) \times \Gamma(TM) \to \Gamma(TM), satisfying the following:

  • It is C^\infty-linear in the first \Gamma(TM)
  • it is \mathbb{R}-linear in the second \Gamma(TM)
  • It satisfies the following relation called the Leibniz rule:

\nabla_X(fv) = (Xf) (v) f \nabla_X(v)

where f is a scalar function on the manifold and fv denotes scalar multiplication of v by f.

Generalizations

The notion of linear connection can be generalized to the more general notion of a connection, where the second TM is replaced by an arbitrary vector bundle E over M.

Operations on a linear connection

Torsion of a linear connection

Further information: torsion of a linear connection

The torsion of a linear connection \nabla is denoted as \tau(\nabla). It is a (1,2)-tensor defined as:

\tau(\nabla)(X,Y)= \nabla_XY - \nabla_YX - [X,Y].

A connection whose torsion is zero is termed a torsion-free linear connection.

Note that torsion makes sense only for linear connections.

Curvature of a linear connection

Further information: Riemann curvature tensor

The curvature of a linear connection \nabla is denoted as R_\nabla. It is defined as:

R_\nabla(X,Y) = \nabla_X\nabla_Y - \nabla_Y\nabla_X - \nabla_{[X,Y]}

The notion of curvature actually makes sense for any connection.