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Notes on Difierential Geometry - CMU

NotesonDi erentialGeometrywithspecialemphasisonsur facesinR3 MarkusDesernoMay 3, 2004 Department of ChemistryandBiochemistry, UCLA,LosAngeles,CA90095-1569,USAMax-Plan ck-Institutf ur Polymerforschung,Ackermannweg 10,55128 Mainz,GermanyThesenotesareanattemptto summarizesomeof thekeymathe-maticalaspectsof di erentialgeometry, as theyapplyin particularto thegeometryof surfacesinR3. Thefocusis notonmathematicalrigorbutratheroncollect ingsomebitsandpiecesof theverypow-erfulmachineryof manifoldsand\post-Newtoniancalculus".Eve nthoughtheultimategoalof eleganceis a completecoordinatefreedescription,thisgo alis farfrombeingachievedhere|notbecausesuch a descriptiondoes notexistyet,butbecausetheauthoris farto thegeometricaspectsaretakenfromFrankel's book[9],onwhich thesenotesrelyheavily. For \classical"di erentialgeometryof curves andsurfacesKreyszigbook[14] hasalsobeentakenas a presentationvariesquitea greatdetail,othersareonlytoucheduponquic kly, mostlywiththeintent to indicateintowhich directiona particularsubjectmight be thetheory of nitions.

Some applications to problems involving the flrst area variation 26 ... isomorphism between these two difierent vector spaces. The determinant of the flrst fundamental form is given by g := detg · jgj · jgijj = 1 2 "ik"jlg ijgkl; (1.6) where "ik is the …

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