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(a)
We have
(b)
(c)
Because  ,  , and  are linearly independent, they form a basis for  . Every 1-form can be written as a linear combinations of , , and , or  . Because  is left-invariant, we have
or
Because , , and are linearly independent, then the coefficients of the previous equation must all be zero. Thus
We know that s is a specific motion in M, and this implies that  and  must all be constants. Thus we have found all 1-forms that are invariant under left translations.
Problem 19
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