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.
The metric or flrst fundamental form on the surface Sis deflned as gij:= ei ¢ej: (1.3) It is a second rank tensor and it is evidently symmetric. If it is furthermore (everywhere) diagonal, the coordinates are called locally orthogonal. ... The modern notation simply calls ...
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