define a linear connection on the tangent bundle or on a vector bundle - Maple Programming Help

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Tensor[Connection] - define a linear connection on the tangent bundle or on a vector bundle

Calling Sequences

Connection(C)

Parameters

C    - the components of the connection to be defined, entered as a type (1, 2) tensor

Description

 • Let $M$ be a manifold and let $\mathrm{χ}\left(M\right)$ be the module (over the ring ${C}^{\infty }\left(M\right)$ of all smooth functions on $M$) of vector fields on. Then a linear connection $\nabla$ on the tangent bundle of $M$ is a mapping which is ${C}^{\infty }\left(M\right)$ linear in its first argument and a derivation on its second argument. If vector fields  define a local frame on, then the coefficients  of with respect to this frame are defined by

Specifying these coefficients is equivalent to defining the connection $\nabla$.

 • More generally, let $E\to M$ be a vector bundle and let $\mathrm{Σ}\left(M\right)$ be the module (over ${C}^{\infty }\left(M\right)$) of sections of $E$. Then a connection on $E$ is a mapping which is linear in its first argument and a derivation on it second argument. If vector fields  define a local frame on $M$ and  define a local basis for the sections of $E$, then the coefficients  of $\nabla$ with respect to these frames are defined by

 • Within the DifferentialGeometry package, connections are displayed using the tensor notation  or ${\mathrm{Γ}}_{\mathrm{bi}}^{a}{\mathrm{η}}^{b}{Z}_{a}{\mathrm{ω}}^{i}$, where the are the dual coframe to the ${X}_{i}$ and the ${\mathrm{η}}^{b}$ are the dual coframe to the ${Z}_{a}$.
 • This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form Connection(...) only after executing the command with(DifferentialGeometry) and with(Tensor) in that order. It can always be used in the long form DifferentialGeometry:-Tensor:-Connection.

Examples

 > with(DifferentialGeometry): with(Tensor):

Example 1.

Create a 2 dimensional manifold $M$ and define a connection on the tangent space of $M$.

 > DGsetup([x, y], M);
 ${\mathrm{frame name: M}}$ (2.1)
 > C := Connection(x*D_x &t dx &t dy - y^2*D_x &t dy &t dy);
 ${C}{:=}{x}{}{\mathrm{D_x}}{}{\mathrm{dx}}{}{\mathrm{dy}}{-}{{y}}^{{2}}{}{\mathrm{D_x}}{}{\mathrm{dy}}{}{\mathrm{dy}}$ (2.2)
 M > Tools:-DGinfo(C, "ObjectType");
 ${"connection"}$ (2.3)

Example 2.

Define a frame on $M$ and use this frame to specify a connection on the tangent space of $M.$

 > FR := FrameData([dx/y, dy/x], M1);
 ${\mathrm{FR}}{:=}\left[{d}{}{\mathrm{Θ1}}{=}\frac{{x}{}{\mathrm{Θ1}}{}{\bigwedge }{}{\mathrm{Θ2}}}{{y}}{,}{d}{}{\mathrm{Θ2}}{=}{-}\frac{{y}{}{\mathrm{Θ1}}{}{\bigwedge }{}{\mathrm{Θ2}}}{{x}}\right]$ (2.4)
 > DGsetup(FR);
 ${\mathrm{frame name: M1}}$ (2.5)
 > C := Connection(E2 &t Theta1 &t Theta2);
 ${C}{:=}{\mathrm{E2}}{}{\mathrm{Θ1}}{}{\mathrm{Θ2}}$ (2.6)

Example 3.

Create a rank 3 vector bundle $E$ on $M$ and define a connection on $E$.

 > DGsetup([x, y], [u, v, w], E);
 ${\mathrm{frame name: E}}$ (2.7)
 > C := Connection(x*D_v &t du &t dy - y*D_u &t dv &t dx);
 ${C}{:=}{-}{y}{}{\mathrm{D_u}}{}{\mathrm{dv}}{}{\mathrm{dx}}{+}{x}{}{\mathrm{D_v}}{}{\mathrm{du}}{}{\mathrm{dy}}$ (2.8)