PullbackVector(Phi, Y, S, freevar)

Example 1.

Suppose Phi: M -> N is an imbedding and Y is a vector field on N which is tangent to the image of M.  Then there exists a unique vector field X on M such that Phi_*(X) = Y; and X can be found using the PullbackVector command. For example, the vectors Y1 and Y2  defined below are both tangent to the unit 3-sphere x^2 + y^2 + z^2 + w^2 = 1 and therefore can be pulled-back by the stereographic projection map Phi1 to the 3-dimensional Euclidean space E3 with coordinates [r, s, t].

We remark that since the vector fields X1 and X2 are uniquely determined, the Lie bracket relations are preserved.

Example 2.

In the following example the map Phi2 is not a local immersion. We can use the freevariable option to specify the name of the indexed variable that will be used to parameterize the vectors X2 such that Phi2_*(X2) = Y2.

We can use the optional third argument to force the vector to belong to a given subspace.

DifferentialGeometry, Pushforward, Transformation