Tensor[TorsionTensor] - calculate the torsion tensor for a linear connection on the tangent bundle
C - a connection on the tangent bundle to a manifold
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=∇XY−∇Y X−X,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.
First create a 2 dimensional manifold M and define a connection on the tangent space of M.
frame name: M
C1 ≔ Connection⁡a⁢D_x &t dx &t dy+b⁢D_x &t dy &t dy
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.
FR ≔ FrameData⁡1⁢dxy,1⁢dyx,M1:
frame name: M1
C2 ≔ Connection⁡E1 &t Θ2 &t Θ2
DifferentialGeometry, Tensor, Christoffel, Physics[Christoffel], Connection, CovariantDerivative, Physics[D_], DirectionalCovariantDerivative
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