Tensor[Connection] - define a linear connection on the tangent bundle or on a vector bundle
C - the components of the connection to be defined, entered as a type (1, 2) tensor
Let M be a manifold and let χM be the module (over the ring C∞M 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 C∞M 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
Specifying these coefficients is equivalent to defining the connection ∇.
More generally, let E→M be a vector bundle and let ΣM be the module (over C∞M) of sections of E. Then a connection ∇ on E is a mapping χM×ΣM→ΣM 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
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.
Create a 2 dimensional manifold M and define a connection on the tangent space of M.
DGsetup([x, y], M);
frame name: M
C := Connection(x*D_x &t dx &t dy - y^2*D_x &t dy &t dy);
Define a frame on M and use this frame to specify a connection on the tangent space of M.
FR := FrameData([dx/y, dy/x], M1);
frame name: M1
C := Connection(E2 &t Theta1 &t Theta2);
Create a rank 3 vector bundle E on M and define a connection on E.
DGsetup([x, y], [u, v, w], E);
frame name: E
C := Connection(x*D_v &t du &t dy - y*D_u &t dv &t dx);
DifferentialGeometry, Tensor, Christoffel, Physics[Christoffel], CovariantDerivative
, Physics[D_], CurvatureTensor
, Physics[Riemann], DirectionalCovariantDerivative, GeodesicEquations, DGinfo, ParallelTransportEquations
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