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

Examples

See Also

Description

• 

Let M be a manifold and let χM be the module (over the ring CM of all smooth functions on M) of vector fields on M. Then a linear connection  on the tangent bundle of M is a mapping χM× χM χM which is CM linear in its first argument and a derivation on its second argument. If vector fields X1, X2, ..., Xn define a local frame on M, then the coefficients Γ  ijk of  with respect to this frame are defined by

XiXj=Γ jikXk.

Specifying these coefficients is equivalent to defining the connection .

• 

More generally, let EM be a vector bundle and let ΣM be the module (over CM) of sections of E. Then a connection  on E is a mapping χM×ΣMΣM which is linear in its first argument and a derivation on it second argument. If vector fields X1, X2, ..., Xn define a local frame on M and Z1,Z2, ...,Zr define a local basis for the sections of E, then the coefficients Γ bia of  with respect to these frames are defined by

XiZb=Γ biaZa.

• 

Within the DifferentialGeometry package, connections are displayed using the tensor notation ΓjikωjXkωi  or ΓbiaηbZaωi, where the ωj are the dual coframe to the Xi and the ηb are the dual coframe to the Za.

• 

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

withDifferentialGeometry:withTensor:

 

Example 1.

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

DGsetupx,y,M

frame name: M

(2.1)

CConnectionxD_x &t dx &t dyy2D_x &t dy &t dy

C:=xD_xdxdyy2D_xdydy

(2.2)
M > 

Tools:-DGinfoC,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.

FRFrameDatadxy,dyx,M1

FR:=dΘ1=xΘ1Θ2y,dΘ2=yΘ1Θ2x

(2.4)

DGsetupFR

frame name: M1

(2.5)

CConnectionE2 &t Θ1 &t Θ2

C:=E2Θ1Θ2

(2.6)

 

Example 3.

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

DGsetupx,y,u,v,w,E

frame name: E

(2.7)

CConnectionxD_v &t du &t dyyD_u &t dv &t dx

C:=yD_udvdx+xD_vdudy

(2.8)

See Also

DifferentialGeometry, Tensor, Christoffel, Physics[Christoffel], CovariantDerivative , Physics[D_], CurvatureTensor , Physics[Riemann], DirectionalCovariantDerivative, GeodesicEquations, DGinfo, ParallelTransportEquations


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