On page 189 of John Lee’s Introduction to Smooth Manifolds it is stated that the set of all smooth left-invariant vector fields on a Lie group $G$ is a linear subspace of the space of all smooth vector fields on $G$. If this set is to be a subspace, the zero vector field should be one of its members. And, this is exactly my question:
Why the zero vector field on $G$ is left-invariant?
Definition of zero vector field is given on p. 177 of that book.
Thanks.
The action of $G$ on the left of a vector field is by linear maps. Any linear map sends the zero vector to the zero vector, and hence the zero vector field to the zero vector field. Therefore, any left action of $G$ on the zero vector field leaves this invariant.