# Difference between revisions of "Transport along a curve"

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− | <math>\frac{D (\phi_t(v)}{dt} = 0</math> | + | <math>\frac{D (\phi_t(v))}{dt} = 0</math> |

Intuitively, we define a rule for ''moving'' the fibre of <math>E</math>, in a manner that is parallel to itself with respect to the connection. | Intuitively, we define a rule for ''moving'' the fibre of <math>E</math>, in a manner that is parallel to itself with respect to the connection. |

## Latest revision as of 18:02, 6 January 2012

## Definition

Let be a differential manifold, a vector bundle on . Let be a smooth curve in . Let denote a connection along of . The transport along defined by maps to the space of sections of along , denoted in symbols as:

such that for any vector :

and

Intuitively, we define a rule for *moving* the fibre of , in a manner that is parallel to itself with respect to the connection.

## Facts

### Connection gives transport

We saw above that a connection along a curve defines a rule of transport along that curve. Consider now a connection on the vector bundle. Then, for every curve, this gives rise to a connection along that curve, and hence a transport along that curve. Hence a connection gives a global rule for transporting frames.