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## Main Question or Discussion Point

When I'm going from a smooth manifold to R with V* X V -> R what does the R scalar stand for. Is it some length in the manifold? and Does this have to do with the way V* and V are defined, since one is a contra-variant and one is a co-variant, are they related in the way the Pythagoras formula is related ? c^2 = a^2 + b^2