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Tensor[CurvatureTensor] - calculate the curvature tensor of a linear connection on the tangent bundle or on a vector bundle

Calling Sequences

    CurvatureTensor(g)  

    CurvatureTensor(C)

Parameters

   g    - a metric on the tangent bundle of a manifold

   C    - a connection on the tangent bundle of a manifold or on a vector bundle

 

Description

Examples

See also

Description

• 

Let  be a connection on the tangent bundle of a manifold M and let χM be the module of all vector fields on M. The curvature tensor of  is the type 1,3 tensor R (contravariant rank 1, covariant rank 3) which, when viewed as a linear map R:χM×χMLχM, is given by

 

RX,YZ = XYZYXZX,YZ.

Here X, Y, and Z are vector fields on M and L denotes the set of linear mappings L:χMχM.

• 

More generally, let  be a connection on a vector bundle EM and let ΣE be the module of all sections of E. The curvature tensor of , when viewed as a linear map Ω:χM×χMLΣE, is given by

ΩX,Yσ = XYσYXσX,Yσ

 Here X and Y are vector fields on M and σΣE is a section of E.

• 

 The first calling sequence computes the Christoffel symbol of the input metric g and returns the Riemann curvature tensor on the tangent bundle as a tensor of type 13. The index type of the output is ["con_bas", "cov_bas", "cov_bas", "cov_bas"]. See TensorIndexType.

• 

The second calling sequence returns the curvature tensor of the input connection C. The index type of the output is ["con_vrt", "cov_vrt", "cov_bas", "cov_bas"]. See TensorIndexType.

• 

The first Bianchi identity for the curvature tensor of a connection on the tangent bundle of a manifold asserts that

 R + S + S·S skewsymmetrized on indices 1,3,4=0

  Here S is the torsion tensor of the connection, and S·S is the contraction of SS over the 3rd and 4th indices.

• 

The second Bianchi Identity asserts that R + R·S skewsymmetrized on indices 3,4,5 = 0. Here R·S denotes the contraction of RS on the third and fifth indices.

• 

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

Examples

withDifferentialGeometry:withTensor:

 

Example 1.

First create a 3 dimensional manifold M and define a metric on the tangent space of M.

DGsetupx,y,z,M

frame name: M

(2.1)
M > 

g1evalDGⅇydx &t dx+dy &t dy+dz &t dz

g1:=ⅇydxdx+dydy+dzdz

(2.2)

 

Calculate the curvature tensor for the metric g1.

M > 

R1CurvatureTensorg1

R1:=14D_xdydxdy+14D_xdydydx+ⅇy4D_ydxdxdyⅇy4D_ydxdydx

(2.3)

 

 

Example 2.

First create a 3 dimensional manifold M and define a connection on the tangent space of M.

M > 

DGsetupx,y,z,M

frame name: M

(2.4)
M > 

C2Connectionx2D_x &t dx &t dyy2D_x &t dy &t dy+yzD_x &t dz &t dy

C2:=x2D_xdxdyy2D_xdydy+yzD_xdzdy

(2.5)
M > 

R2CurvatureTensorC2

R2:=2xD_xdxdxdy2xD_xdxdydxyD_xdzdydz+yD_xdzdzdy

(2.6)

 

Here are two simple procedures we shall use for checking the Bianchi identities.

Bianchi1 := proc(R, C)

local S,T1,T2,T3,T4

STorsionTensorC

T1R

T2CovariantDerivativeS,C:

T3ContractIndicesS,S,3,1

T4evalDGT1T2+T3

SymmetrizeIndicesT4,2,3,4,SkewSymmetric

M > 

end:

 

Bianchi2 := proc(R, C)

local T1,T2,S

STorsionTensorC

T1CovariantDerivativeR,C

T2ContractIndicesR,S,3,1

SymmetrizeIndicesT1 &plus T2,3,4,5,SkewSymmetric

M1 > 

end:

 

Use the above programs to check the Bianchi identities.

M > 

Bianchi1R2,C2

0D_xdxdxdx

(2.7)
M > 

Bianchi2R2,C2

0D_xdxdxdxdx

(2.8)

 

Example 3.

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

M > 

DGsetupx,y,z,M

frame name: M

(2.9)
M > 

FRFrameDatax2dxy,zdyx,xydz,M1:

M > 

DGsetupFR

frame name: M1

(2.10)
M1 > 

C3ConnectionE2 &t Θ1 &t Θ2

C3:=E2Θ1Θ2

(2.11)
M1 > 

R3CurvatureTensorC3

R3:=yx3E2Θ1Θ1Θ2+yx3E2Θ1Θ2Θ11zxyE2Θ1Θ2Θ3+1zxyE2Θ1Θ3Θ2

(2.12)

 

Use the above programs to check the Bianchi identities.

M1 > 

Bianchi1R3,C3

0E1Θ1Θ1Θ1

(2.13)
M1 > 

Bianchi2R3,C3

0E1Θ1Θ1Θ1Θ1

(2.14)

 

Example 4.

First create a rank 3 vector bundle EM and define a connection on E.

M1 > 

DGsetupx,y,z,u,v,w,E

frame name: E

(2.15)
E > 

C4ConnectionxzD_v &t du &t dy

C4:=xzD_vdudy

(2.16)
E > 

R4CurvatureTensorC4

R4:=zD_vdudxdyzD_vdudydxxD_vdudydz+xD_vdudzdy

(2.17)

See also

Physics[Riemann], DifferentialGeometry, Tensor, Christoffel, Physics[Christoffel], Connection, CovariantDerivative, Physics[D_], SectionalCurvature, RicciScalar, RicciTensor, Physics[Ricci], TorsionTensor, Physics[Weyl]


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