Khudaverdian - Introduction to Geometry, Topologia i Geometria

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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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