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...Lecture riemannian metrics this introduces which dene lengths of vectors and curves in the manifold denition there is one more crucial ingredient we need to introduce for dealing with manifolds angles given a smooth since know what it means curve be have well dened notion tangent vector all order distance on way dening speed that length its metric g m smoothly chosen inner product gx txm r each spaces inother words x satises u v ifand only if furthermore sense any elds xandy function y issmooth locally can described terms coecients local chart by smoothness equivalent ij i j coecient functions some example standard euclidean space special rn made many ways let f abounded n then c suciently large ji arepositivedeniteeverywhere andsodeneariemannianmetric existence every carries fact them will prove using an argument very similar used showing connections choose atlas subordinate partition unity oneach regions as d clearly symmetric iu so denes bea piecewise b lof l t dt two points p q wed...