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# Let v , w, and y be dierentiable vector elds in U S, f : U R be dierentiable functions in S , y (f ) be the derivative of f in the direction of y and...

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Let v , w , and y be diﬀerentiable vector ﬁelds in U S,f : U R be diﬀeren- tiable functions in S , y ( f ) be the derivative of f in the direction of y and a , b be real numbers. Show that the covariant derivative has the following property y ( < v,w > ) = < D y v,w > + < v,D y w > 1

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Let M be any Riemannian manifold (in particular surface in R3 ), and let
for any smooth curve c and any pair of parallel vector ﬁelds P , P , along c,
we have &lt; P, P &gt;= constant. If you...

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