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DifferentialGeometry

  

FrameData

  

calculate the structure equations for a generic (anholonomic) frame

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

FrameData(Fr, FrName)

Parameters

Fr

-

a list of vector fields or differential 1-forms on a manifold M

FrName

-

an unassigned Maple name or string, the name to that will be assigned to the frame with the command DGsetup

Description

• 

There are many situations in differential geometry where computations are tremendously simplified by using a frame or co-frame other than the standard coordinate frame or co-frame. We refer to a general (local) frame or co-frame on a manifold as an anholonomic frame.  Important examples of anholonomic frames are: the orthogonal frames constructed from a metric on a manifold; the null frames used in general relativity; the left (or right) invariant vector fields on a Lie group; the moving frames adapted to a free group action on a manifold.  Cartan's method of equivalence provides an algorithmic approach to constructing adapted co-frames for Pfaffian systems.

• 

All of the commands in the DifferentialGeometry package and the Tensor subpackage work with general anholonomic frames.  At present, the commands in the JetCalculus package work only with the standard coordinate frame.

• 

The structure equations of a frame Fr = [X_1, X_2, ...] (the X_i are vector fields) are the Lie bracket relations

[X_i, X_j] = F_{ij}^k X_k    (sum on k).

• 

The structure equations for a co-frame Omega = [omega^1,  omega^2, ...] (the omega^k are differential 1-forms) are the exterior derivative formulas

d(omega^k) = G_{ij}^k omega ^i &w omega ^j.

If the co-frame Omega is the dual co-frame to the frame Fr, then the structure functions are related by G_{ij}^k = -1/2*F_{ij}^k).

• 

To work in DifferentialGeometry with anholonomic frames on a manifold M, first define an underlying coordinate system on M and define the anholonomic frame or co-frame relative to this coordinate system. Use the command FrameData to generate the structure equations for this frame (along with other data).  Pass the results of the FrameData procedure to the DGsetup procedure.

• 

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

Examples

withDifferentialGeometry:withTensor:

 

Example 1.

Calculate the structure equations for the frame Fr.  Create a manifold N with local coordinate (x, y) and Fr as the frame for the tangent bundle.

DGsetupx,y,M:

FrevalDGxD_x+yD_y,y2D_x+x2D_y

FrxD_x+yD_y,y2D_x+x2D_y

(1)

FDFrameDataFr,N

FDE1,E2=E2

(2)

DGsetupFD,verbose

The following coordinates have been protected:

x,y

The following vector fields have been defined and protected:

E1,E2

The following differential 1-forms have been defined and protected:

Θ1,Θ2

frame name: N

(3)

Calculate the exterior derivative of the function x*y in terms of the co-frame dual to Fr.

ExteriorDerivativexy

2xyΘ1+x3+y3Θ2

(4)

 

Example 2.

Find an orthonormal co-frame for the metric g. Use this co-frame to compute the curvature tensor and its first covariant derivative.

DGsetupx,y,M:

gevalDGdx &t dx+ⅇ2xdy &t dy

gdxdx+ⅇ2xdydy

(5)

ΩevalDGdx,ⅇxdy

Ωdx,ⅇxdy

(6)

coFDFrameDataΩ,N

coFDdΘ1=0,dΘ2=Θ1Θ2

(7)

DGsetupcoFD,E,ω,verbose

The following coordinates have been protected:

x,y

The following vector fields have been defined and protected:

E1,E2

The following differential 1-forms have been defined and protected:

ω1,ω2

frame name: N

(8)

gevalDGω1 &t ω1+ω2 &t ω2

gω1ω1+ω2ω2

(9)

CChristoffelg

CE1ω2ω2+E2ω1ω2

(10)

RCurvatureTensorC

RE1ω2ω1ω2+E1ω2ω2ω1+E2ω1ω1ω2E2ω1ω2ω1

(11)

R1CovariantDerivativeR,C

R10E1ω1ω1ω1ω1

(12)

 

Example 3.

In this example we shall encode the Liouville equation u_xy = exp(u) as a exterior differential system on a 7 manifold N with a co-frame adapted to the hyperbolic structure of the equation. The steps are:

1. Create a manifold M with coordinates (x, y, u, p, q, r, t) -- here we are using the classical notation for derivatives p = u_x, q = u_y, r = u_xx, t = u_yy.

2. Define a co-frame Omega on M by Omega = [du - p*dx - q*dy, dp - r*dx - exp(u)*dy, dq - exp(u)*dx - t*dy, dx, dr - p*dp, dy, dt - q*dq].

3. Compute the structure equations for the co-frame Omega using the FrameData command.

4. Initialize the manifold N with the co-frame Omega.  Label the first 3 elements of the co-frame on N as theta1, theta2, theta3, and the last 4 elements as pi1, pi2, pi3, pi4.

5. Compute the exterior derivatives of theta1, theta2, theta3.

withDifferentialGeometry:

DGsetupx,y,u,p,q,r,t,M:

ΩevalDGdupdxqdy,dprdxⅇudy,dqⅇudxtdy,dx,drpdp,dy,dtqdq

Ωpdxqdy+du,rdxⅇudy+dp,ⅇudxtdy+dq,dx,pdp+dr,dy,qdq+dt

(13)

coFDFrameDataΩ,N

coFDdΘ1=Θ2Θ4Θ3Θ6,dΘ2=ⅇuΘ1Θ6pΘ2Θ4+Θ4Θ5,dΘ3=ⅇuΘ1Θ4qΘ3Θ6+Θ6Θ7,dΘ4=0,dΘ5=0,dΘ6=0,dΘ7=0

(14)

DGsetupcoFD,E,θ1,θ2,θ3,π1,π2,π3,π4

frame name: N

(15)

ExteriorDerivativeθ1

θ2π1θ3π3

(16)

ExteriorDerivativeθ2

ⅇuθ1π3pθ2π1+π1π2

(17)

ExteriorDerivativeθ3

ⅇuθ1π1qθ3π3+π3π4

(18)
N > 

See Also

CovariantDerivative

CurvatureTensor

DGsetup

DifferentialGeometry

DualBasis

Tensor

 


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