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...Basic geometry of submanifolds wernerballmann contents introduction notation intrinsic first fundamental form distance variation arc length and geodesics parallel vector elds extrinsic hypersurfaces more examples some exercises acknowledgments references date last update n i discuss the in r which motivated most results techniques present day dierential this is not a historical account with today s point view explain main insights into before time gauss riemann there are two aspects our setting describes inside only role ambient space rn to induce way measuring angles lengths geometric objects contained deals shape as subsets all considerations local arguments go through for immersions well since many elementary come chapter following m manifold dimension f an immersion data pair assume that reader familiar theory curves also familiarity manifolds who on good terms yet advised think them open or euclidean spaces if u v real nite smooth function then partial derivative respect xi denote...