Parallel Transport

PARALLEL FIELDS ON SMOOTH CURVES

A Vector Field

X

\gamma

is called ''parallel'' if

:

abla_{\dot\gamma(t)}X=0

for any ''t''.

PARALLEL TRANSPORT

Let ''M'' be a smooth manifold with connection

abla

\gamma: I 	o M

a smooth curve parameterized by the open interval ''I'' which includes 0 and let

X_0 \in \mathrm{T}_{\gamma(0)} M

be a tangent vector at γ(0). Then there exists a unique vector field ''X'' along

\gamma

such that

abla_{\dot{\gamma}} X = 0

and

X(0) = X_0

. The vector field ''X'' is called the parallel transport of

X_0

along

\gamma

.

GEODESICS

Geodesics on ( Pseudo -) Riemannian Manifolds are defined as follows. Let ''M'' be a smooth manifold with connection

abla

. A smooth curve

\gamma: I \longrightarrow M

is a geodesic if

\dot\gamma

(as a vector field along

\gamma

) is parallel along itself. In other words, if

:

abla_{\dot\gamma(t)}\dot\gamma = 0

PARALLEL AND GEODESIC VECTOR FIELDS

A vector field

X

on ''M'' is called ''parallel'' if

:

abla_Y X = 0

orall Y \in \mathrm{T}M

and ''geodesic'' if

:

abla_X X = 0

.

SEE ALSO

	CATEGORIES ABOUT PARALLEL TRANSPORT

	riemannian geometry

	connection mathematics

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