Looks ok to me.knowwhatyoudontknow said:OK. Thanks. If I have γ(t) = exp(tv) so that γ(0) = p and γ'(0) = v and Lg is a left action then is the following correct?
Lg*γ'(0) = (Lg o γ(t))'(0)
= d/dt|t=0Lg(γ(t))
= dLg/dt|t=0.v
= gv
In other words dLg/dt|t=0 ≡ g
The pushforward of a vector field is a way of transforming a vector field from one space to another. It is a mathematical operation that takes a vector field defined on one manifold and maps it to a vector field on another manifold.
The pushforward of a vector field is calculated using the differential of a smooth map between the two manifolds. This differential is a linear map that takes tangent vectors from one manifold to tangent vectors on the other manifold, and this is what transforms the vector field.
The pullback of a vector field is the inverse operation of the pushforward. It takes a vector field on one manifold and maps it to a vector field on another manifold. It is also known as the covariant derivative or the pushforward of a vector field.
The pullback of a vector field is the inverse operation of the pushforward. This means that if we apply the pushforward to a vector field and then apply the pullback, we will get back the original vector field. In other words, the pullback "undoes" the transformation done by the pushforward.
The pushforward and pullback of vector fields are important tools in differential geometry and have applications in many fields, including physics, engineering, and computer graphics. They are used to transform vector fields in different coordinate systems, which is essential in understanding the behavior of physical systems and creating accurate simulations.