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...Differential geometry lectures by p m h wilson notesbydavidmehrle dfm cam ac uk cambridgeuniversity mathematicaltripospartiii michaelmas contents lecture last updated april october administrative stuff there are some notes online they have that we won t cover the best book is spivak manifoldsandvectorspaces smoothmanifolds denition if u rm and n r say smooth or c has continuous partial derivatives of all orders a topological space x called second countable exists collection b open subsets such any subset maybewrittenastheunionofsetsofb hausdorff manifold dimension d each point an neighbor hood nbhd homeomorphictoanopensubsetu ofrd byahomeomorphism puqrd thepairpu qofahomeomorphismandopensubsetofmiscalledachart given v with xv charts pu q pv puq rd pvq homeomorphism puxvqn puxvqofopensubsets vu chart call coordinate functions where i projection onto th coordinates structure on for tu au ii xu transition function issmooth thecharts aresaidtobecompatible iii maximal respect to in particul...