Parallel transport

Parallel transport of a vector on the spherical surface along a closed path from A to N and B and back to A. The angle by which the vector is rotated is proportional to the enclosed area within the path. ${\displaystyle \alpha }$

In the differential geometry referred parallel transport ( English transport parallel or parallel translation ) or parallel displacement of a method geometric objects along smooth curves in a manifold to transport. In 1917, Tullio Levi-Civita expanded Riemannian geometry to include this term, which then led to the definition of the context .

If the manifold has a covariant derivative (in the tangential bundle ), then vectors in the manifold can be transported along curves in such a way that they remain parallel with regard to the relationship belonging to the covariant derivative . Correspondingly, a parallel transport can be constructed for each context. A Cartan context even allows curves to be lifted from the manifold into the associated principal bundle . Such a curve lifting allows the parallel transport of reference systems, i.e. the transport of a base from one point to another. The parallel transport belonging to a connection thus allows in a certain way to move the local geometry of a manifold along a curve.

Just as a parallel transport can be constructed from a connection, conversely, a connection can be constructed from a parallel transport. In this respect, a connection is an infinitesimal analogue to a parallel transport or a parallel transport is the local realization of a connection. In addition to the local realization of a connection, a parallel transport also provides a local realization of the curvature , the holonomy . The set of Ambrose-Singer makes this relationship between curvature and holonomy explicitly.

Parallel vector field

Let be an interval and a differentiable manifold with connection . ${\displaystyle I\subseteq \mathbb {R} }$ ${\displaystyle M}$ ${\displaystyle \nabla }$

A vector field along a curve is called parallel along if applies to all . ${\displaystyle V\colon \ M\to T_{\gamma (t)}M}$ ${\displaystyle \gamma \colon \ I\to M}$ ${\displaystyle \gamma }$ ${\displaystyle \nabla _{\gamma '(t)}V(\gamma (t))=0}$ ${\displaystyle t}$

A vector field is said to be parallel if it is parallel with respect to every curve in the manifold.

Parallel transport

Let be a differentiable manifold, a curve and two real numbers. Then for each there is a unique parallel vector field along , so that holds. With the help of this statement of existence and uniqueness, one can define the mapping, which is called parallel transport : the mapping ${\displaystyle M}$ ${\displaystyle \gamma \colon \ I\to M}$ ${\displaystyle t_{0},t_{1}\in I}$ ${\displaystyle v_{0}\in T_{\gamma (t_{0})}M}$ ${\displaystyle V\colon \ M\to T_{\gamma (t)}M}$ ${\displaystyle \gamma }$ ${\displaystyle v_{0}=V(\gamma (t_{0}))}$

${\displaystyle {\begin{aligned}P_{\gamma (t_{0}),\gamma (t_{1})}\colon \ T_{\gamma (t_{0})}M\to T_{\gamma (t_{1})}M,\quad v_{0}\mapsto V(\gamma (t_{1})),\end{aligned}}}$

which assigns its unique parallel vector field evaluated at the point to a vector . ${\displaystyle v_{0}\in T_{\gamma (t_{0})}M}$ ${\displaystyle V}$ ${\displaystyle \gamma (t_{1})}$

The existence and uniqueness follow from the property of initial value problems of linear ordinary differential equation systems , the unique solution of which exists globally for all times according to the global version of Picard-Lindelof's theorem .

For the Levi-Civita context

The most important special case for the parallel transport is the transport of a tangential vector along a curve on a Riemannian manifold , the relationship being the Levi-Civita relationship .

In concrete terms: If a tangential vector is at the point and a smooth curve with , then a vector field is called along , i.e. H. with , and if and only then, parallel transport of , if: ${\displaystyle v_{0}\in T_{p}M}$ ${\displaystyle p}$ ${\displaystyle \gamma (t)}$ ${\displaystyle \gamma (0)=p}$ ${\displaystyle v(t)}$ ${\displaystyle \gamma }$ ${\displaystyle v(t)\in T_{\gamma (t)}M}$ ${\displaystyle v_{0}}$

1. ${\displaystyle \nabla _{\gamma '(t)}v(t)=0}$
2. ${\displaystyle v(0)=v_{0},}$

so if the covariant derivative of along vanishes. ${\displaystyle v(t)}$ ${\displaystyle \gamma }$

This is a linear initial value problem of the first order, of which the existence and uniqueness of a solution can be shown (see above).

The amount of a vector that is shifted in parallel is constant:

${\displaystyle {\frac {d}{dt}}g(v(t),v(t))=2g(\nabla _{\gamma '(t)}v(t),v(t))=2\cdot 0=0.}$

Along a geodesic

In the case of a geodesic , the parallel transport has special properties. ${\displaystyle \gamma }$

For example, the tangential vector of a geodetic parameterized proportionally to the arc length is itself parallel:

${\displaystyle \nabla _{\gamma '(t)}\gamma '(t)=0}$

Because this was exactly the definition of a geodesic on a Riemannian manifold.

The angle between the tangential vector of the geodesic and the vector is constant, since the amounts of both vectors are also constant (see above).

${\displaystyle {\frac {d}{dt}}g(v(t),\gamma '(t))=g(\nabla _{\gamma '(t)}v(t),\gamma '(t))+g(v(t),\nabla _{\gamma '(t)}\gamma '(t))=0+0=0.}$

In Euclidean spaces

In Euclidean space , the covariant derivative is the normal derivative in a certain direction. It disappears if , apart from the base point, is constant, i.e. i.e., if all vectors are parallel. ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle v(t)}$ ${\displaystyle v(t)}$

The parallel transport is therefore a generalization of the parallel displacement of a vector along a curve.

literature

• Christian Bär : Elementary Differential Geometry . 2nd Edition. De Gruyter, Berlin / New York 2010, ISBN 978-3-11-022458-0 .
• John M. Lee: Riemannian Manifolds. An Introduction to Curvature . 1st edition. Springer, New York / Berlin / Heidelberg 1997, ISBN 0-387-98322-8 .
• Manfredo do Carmo : Riemannian Geometry . 1st edition. Birkhäuser, Boston / Basel / Berlin 1992, ISBN 0-8176-3490-8 (Portuguese: Elementos de geometria diferencial . Rio de Janeiro 1971. Translated by Francis Flaherty, first edition: Ao Livro Técnico).

Individual evidence

1. ^ Levi-Civita, Tullio . In: Guido Walz (Ed.): Lexicon of Mathematics . 1st edition. Spectrum Academic Publishing House, Mannheim / Heidelberg 2000, ISBN 3-8274-0439-8 .