Transcription of Notes on Difierential Geometry - CMU
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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.
Hence, the components of the inverse metric are given by µ g11 g12 g21 g22 ¶ = 1 g µ g22 ¡g21 ¡g12 g11 ¶: (1.5) By virtue of Eqn. (1.4) the metric tensor can be used to raise and lower indices in tensor equations. Technically, \indices up or down" means that we are referring to components of tensors which live in the tangent space or the
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