Khudaverdian - Introduction to Geometry (lecture notes).pdf

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IntroductiontoGeometry
itisadraftoflecturenotesofH.M.Khudaverdian.
Manchester,18May2010
Contents
1Euclideanspace 3
1.1Vectorspace. ........................... 3
1.2Basicexampleof n -dimensionalvectorspace|R n ....... 4
1.3Lineardependenceofvectors................... 4
1.4Dimensionofvectorspace.Basisinvectorspace. ....... 7
1.5Scalarproduct.Euclideanspace................. 9
1.6OrthonormalbasisinEuclideanspace..............10
1.7Transitionmatrices.Orthogonalmatrices............12
1.8Orthogonal2 £ 2matrices....................15
1.9Orientationinvectorspace....................17
1.10 y LinearoperatorinE 3 preservinigorientationisarotation..23
1.11VectorproductinorientedE 3 ..................24
1.11.1Volumeofparallelepiped ................30
2Di®erentialformsinE 2 andE 3 31
2.1Tangentvectors,curves,velocityvectorsonthecurve.....31
2.2Reparameterisation........................32
2.30-formsand1-forms........................34
2.4Di®erential1-forminarbitrarycoordinates ..........39
2.5Integrationofdi®erential1-formsovercurves..........42
2.6Integralovercurveofexactform.................46
2.7Di®erential2-formsinE 2 .....................47
2.80-forms(functions) d ¡! 1-forms d ¡! 2-forms..........48
1
2.9 y Exactandclosedforms.....................49
2.10 y Integrationoftwo-forms.Areaofthedomain.........50
3CurvesinE 3 .Curvature 51
3.1Curves.Velocityandaccelerationvectors............51
3.2Behaviourofaccelerationvectorunderreparameterisation..53
3.3Lengthofthecurve .......................55
3.4Naturalparameterisationofthecurves.............55
3.5Curvature.CurvatureofcurvesinE 2 ..............58
3.6Curvatureofcurveinanarbitraryparameterisation.......59
4SurfacesinE 3 .CurvaturesandShapeoperator. 62
4.1Coordinatebasis,tangentplanetothesurface..........63
4.2Curvesonsurfaces.Lengthofthecurve.Internalandexternal
pointoftheview.FirstQuadraticForm............63
4.3Unitnormalvectortosurface..................67
4.4 y Curvesonsurfaces|normalaccelerationandnormalcurvature68
4.5Shapeoperatoronthesurface..................70
4.6Principalcurvatures,Gaussianandmeancurvaturesandshape
operator..............................72
4.7 y Principalcurvaturesandnormalcurvature...........74
5 y Appendices 75
5.1Formulaeforvector¯eldsanddi®erentialsincylindricaland
sphericalcoordinates.......................75
5.2Curvatureandsecondordercontact(touching)ofcurves...78
5.3Integralofcurvatureoverplanarcurve. ............80
5.4Relationsbetweenusualcurvaturenormalcurvatureandgeodesic
curvature..............................82
5.5Normalcurvatureofcurvesoncylindersurface.........84
5.6Conceptofparalleltransport...................86
5.7Paralleltransportofvectorstangenttothesphere. ......87
5.8Paralleltransportalongaclosedcurveonarbitrarysurface...89
5.9GaussBonnetTheorem......................90
5.10TheoremaEgregium.......................92
2
1Euclideanspace
Werecallimportantnotionsfromlinearalgebra.
1.1Vectorspace.
Vectorspace V onrealnumbersisasetofvectorswithoperations"+
"|additionofvectorand" ¢ "|multiplicationofvectorLonrealnumber
(sometimescalledcoe±cients,scalars).Theseoperationsobeythefollowing
axioms
²8 a ; b 2V; a+b 2V ,
²8¸2 R ;8 a 2V;¸ a 2V .
²8 a ; ba+b=b+a(commutativity)
²8 a ; b ; c ; a+(b+c)=(a+b)+c(associativity)
²9 0suchthat 8 aa+0=a
²8 athereexistsavector ¡® suchthata+( ¡ a)=0.
²8¸2 R ;¸ (a+b)= ¸ a+ ¸ b
²8¸;¹2 R( ¸ + ¹ )a= ¸ a+ ¹ a
² ( ¸¹ )a= ¸ ( ¹ a)
² 1a=a
Itfollowsfromtheseaxiomsthatinparticularly0isuniqueand ¡ ais
uniquelyde¯nedbya.(Proveit.)
Examplesofvectorspaces...
3
1.2Basicexampleof n -dimensionalvectorspace|R n
Abasicexampleofvectorspace(overrealnumbers)isaspaceofordered
n -tuplesofrealnumbers.
R 2 isaspaceofpairsofrealnumbers.R 2 = f ( x;y ) ;x;y2 R g
R 3 isaspaceoftriplesofrealnumbers.R 3 = f ( x;y;z ) ;x;y;z2 R g
R 4 isaspaceofquadruplesofrealnumbers.R 4 = f ( x;y;z;t ) ;x;y;z;t;2 R g
andsoon...
R n |isaspaceof n -typlesofrealnumbers:
R n = f ( x 1 ;x 2 ;:::;x n ) ;x 1 ;:::;;x n 2 R g (1.1)
Ifx ; y 2 R n aretwovectors,x=( x 1 ;:::;x n ),y=( y 1 ;:::;y n )then
x+y=( x 1 + y 1 ;:::;x n + y n ) :
andmultiplicationonscalarsisde¯nedas
¸ x= ¸¢ ( x 1 ;:::;x n )=( ¸x 1 ;:::;¸x n ) ; ( ¸2 R) :
( ¸2 R).
1.3Lineardependenceofvectors
Weoftenconsiderlinearcombinationsinvectorspace:
X
¸ i x i = ¸ 1 x 1 + ¸ 2 x 2 + ¢¢¢ + ¸ m x m ; (1.2)
i
where ¸ 1 ;¸ 2 ;:::;¸ m arecoe±cients(realnumbers),x 1 ; x 2 ;:::; x m arevectors
fromvectorspace V .
Wesaythatlinearcombination(1.2)is trivial ifallcoe±cients ¸ 1 ;¸ 2 ;:::;¸ m
areequaltozero.
¸ 1 = ¸ 2 = ¢¢¢ = ¸ m =0 :
Wesaythatlinearcombination(1.2)is nottrivial ifatleastoneofcoef-
¯cients ¸ 1 ;¸ 2 ;:::;¸ m isnotequaltozero:
¸ 1 6 =0 ; or ¸ 2 6 =0 ; or ::: or ¸ m 6 =0 :
Recallde¯nitionoflinearlydependentandlinearlyindependentvectors:
4
De¯nitionThevectors f x 1 ; x 2 ;:::; x m g invectorspace V are linearly
dependent ifthereexistsanon-triviallinearcombinationofthesevectors
suchthatitisequaltozero.
Inotherwordswesaythatthevectors f x 1 ; x 2 ;:::; x m g invectorspace V
are linearlydependent ifthereexistcoe±cients ¹ 1 ;¹ 2 ;:::;¹ m suchthatat
leastoneofthesecoe±cientsisnotequaltozeroand
¹ 1 x 1 + ¹ 2 x 2 + ¢¢¢ + ¹ m x m =0 : (1.3)
Respectivelyvectors f x 1 ; x 2 ;:::; x m g are linearlyindependent iftheyare
notlinearlydependent.Thismeansthatanarbitrarylinearcombinationof
thesevectorswhichisequalzeroistrivial.
Inotherwordsvectors f x 1 ; x 2 ; x m g are linearlyindependent ifthecondi-
tion
¹ 1 x 1 + ¹ 2 x 2 + ¢¢¢ + ¹ m x m =0
impliesthat ¹ 1 = ¹ 2 = ¢¢¢ = ¹ m =0.
Veryusefulandworkable
Proposition Vectorsf x 1 ; x 2 ;:::; x m ginvectorspaceVarelinearly
dependentifandonlyifatleastoneofthesevectorsisexpressedvialinear
combinationofothervectors:
x i =
X
¸ j x j : (1.4)
Proof .Ifthecondition(1.4)isobeyedthen x i ¡ P
j6 = i
j6 = i ¸ j x j =0.This
non-triviallinearcombinationisequaltozero.Hencevectors fx 1 ;:::; x m g
arelinearlydependent.
Nowsupposethatvectors f x 1 ;:::; x m g arelinearlydependent.This
meansthatthereexistcoe±cients ¹ 1 ;¹ 2 ;:::;¹ m suchthatatleastoneof
thesecoe±cientsisnotequaltozeroandthesum(1.3)equalstozero.WLOG
supposethat ¹ 1 6 =0.Weseethatto
x 1 = ¡ ¹ 2
¹ 1 x 2 ¡ ¹ 3
¹ 1 x 3 ¡¢¢¢¡ ¹ m
¹ 1 x m ;
i.e.vectorx 1 isexpressedaslinearcombinationofvectors f x 2 ; x 3 ;:::; x m g .
Formulateandgiveaproofofuseful
5
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