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Tensor[TorsionTensor] - calculate the torsion tensor for a linear connection on the tangent bundle

Calling Sequences

     TorsionTensor(C)

Parameters

   C    - a connection on the tangent bundle to a manifold

 

Description

Examples

See Also

Description

• 

Let M be a manifold and let  be a linear connection on the tangent bundle of M. The torsion tensor S of  is the rank 3 tensor (tensor of type12) defined by  SX,Y=XYY XX,Y.  Here X,Y are vector fields on M.

• 

The connection  is said to be symmetric if its torsion tensor S vanishes.

• 

This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form TorsionTensor(...) only after executing the command with(DifferentialGeometry) and with(Tensor) in that order. It can always be used in the long form DifferentialGeometry:-Tensor:-TorsionTensor.

Examples

withDifferentialGeometry:withTensor:

 

Example 1.

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

DGsetupx,y,M

frame name: M

(2.1)
M > 

C1ConnectionaD_x&tdx&tdy+bD_x&tdy&tdy

C1:=aD_xdxdy+bD_xdydy

(2.2)
M > 

TorsionTensorC1

aD_xdxdy+aD_xdydx

(2.3)

 

Example 2.

Define a frame on M and use this frame to specify a connection C2 on the tangent space of M.  While the connection C2 is "symmetric" in its covariant indices, it is not a symmetric connection.

M > 

FRFrameData1ydx,1xdy,M1:

M > 

DGsetupFR

frame name: M1

(2.4)
M1 > 

C2ConnectionE1&tΘ2&tΘ2

C2:=E1Θ2Θ2

(2.5)
M1 > 

TorsionTensorC2

xE1Θ1Θ2yxE1Θ2Θ1yyE2Θ1Θ2x+yE2Θ2Θ1x

(2.6)

See Also

DifferentialGeometry, Tensor, Christoffel, Physics[Christoffel], Connection, CovariantDerivative, Physics[D_], DirectionalCovariantDerivative