Unlike in the case, the exponential mapping is not onto.
2.
The physics convention for Lie algebras and the exponential mapping is used.
3.
The exponential mapping is a polynomial mapping from the Lie algebra to the corresponding subgroup by nilpotence.
4.
The exponential mapping here is not always a surjective map on sections; this can be seen for example when " M " is a punctured disk in the complex plane.
5.
The exponential mapping is ( in this context ), after all, a map from a Lie algebra to the connected group generated by it, not to another isomorphic group or some quotient.
6.
This is a closed loop in, i . e . rotations ranging from 0 to around the z-axis under the exponential mapping, but it is only " half " " a loop in, ending at.
7.
In a smooth manifold setting, the exponential mapping is simply " defined " to be the time one flow of a ( left invariant ) smooth vector field without the strong motivational point of the power series for its name.
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