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Differential Geometry
Lectures by P.M.H. Wilson
NotesbyDavidMehrle
dfm33@cam.ac.uk
CambridgeUniversity
MathematicalTriposPartIII
Michaelmas2015
Contents
Lecture 1 . . . . . . . . . . . . . . 2 Lecture13 . . . . . . . . . . . . . 35
Lecture 2 . . . . . . . . . . . . . . 4 Lecture14 . . . . . . . . . . . . . 38
Lecture 3 . . . . . . . . . . . . . . 6 Lecture15 . . . . . . . . . . . . . 41
Lecture 4 . . . . . . . . . . . . . . 9 Lecture16 . . . . . . . . . . . . . 44
Lecture 5 . . . . . . . . . . . . . . 12 Lecture17 . . . . . . . . . . . . . 48
Lecture 6 . . . . . . . . . . . . . . 15 Lecture18 . . . . . . . . . . . . . 51
Lecture 7 . . . . . . . . . . . . . . 18 Lecture19 . . . . . . . . . . . . . 54
Lecture 8 . . . . . . . . . . . . . . 20 Lecture20 . . . . . . . . . . . . . 57
Lecture 9 . . . . . . . . . . . . . . 23 Lecture21 . . . . . . . . . . . . . 60
Lecture 10 . . . . . . . . . . . . . 26 Lecture22 . . . . . . . . . . . . . 64
Lecture 11 . . . . . . . . . . . . . 29 Lecture23 . . . . . . . . . . . . . 68
Lecture 12 . . . . . . . . . . . . . 32 Lecture24 . . . . . . . . . . . . . 70
Last updated April 1, 2016.
1
Lecture 1 8October2015
Administrative Stuff
There are some Lecture Notes online. They have some stuff that we won’t
cover. The best book is Spivak.
ManifoldsandVectorSpaces
SmoothManifolds
Definition 1. If U Ă Rm and δ: U Ñ R, we say that δ is smooth or C8 if has
continuous partial derivatives of all orders.
Definition 2. A topological space X is called second countable if there exists
a countable collection B of open subsets of X such that any open subset of X
maybewrittenastheunionofsetsofB.
Definition 3. A Hausdorff, second countable topological space X is called
a topological manifold of dimension d if each point has an open neighbor-
hood(nbhd)homeomorphictoanopensubsetU ofRd byahomeomorphism
φ: U „ φpUqĂRd.
ThepairpU,φqofahomeomorphismandopensubsetofMiscalledachart:
given open subsets U and V of X with U XV ‰ H, and charts pU,φ q and
U
pV,φ q, with φ : U Ñ φpUq Ă Rd and φ : V Ñ φpVq Ă Rd, we have a
V U ´1 V
homeomorphism φ “φ ˝φ :φ pUXVqÑφ pUXVqofopensubsets
VU V U U V
of Rd.
Given a chart pU,φ q and a point p P U, we call U a coordinate neighbor-
U
hood of p and we call the functions x : U Ñ R given by π ˝φ (where π is
i i U i
the projection onto the i-th coordinate) coordinates of U.
Definition 4. A smooth structure on a topological manifold is a collection A
of charts pU ,φ q for α P A, such that
α α
(i) tU | α P Au is an open cover of M;
α
(ii) for any α, β P A such that U XU ‰ H, the transition function φ “
α β βα
φ ˝φ´1issmooth. Thechartsφ andφ aresaidtobecompatible;
β α α β
(iii) the collection of charts φ is maximal with respect to (ii). In particular,
α
this means that if a chart φ is compatible with all the φ , then it’s already
α
in the collection.
Remark 5. Since φ “ φ´1: φ pU XU q Ñ φ pU XU q, both φ and φ
αβ βα β α β α α β βα αβ
are in fact diffeomorphisms (since by assumption, they are smooth).
Thisremarkshowsthatitem(ii)inDefinition4impliesthattransitionfunc-
tions are diffeomorphisms.
For notation, we sometimes write U “U XU .
αβ α β
2
Definition 6. A collection of charts tpU ,φ q | α P Au satisfying items (i) and
α α
(ii) in Definition 4 is called an atlas.
Claim7. Anyatlas A is contained in a unique maximal atlas and so defines a
uniquesmoothstructureonthemanifold.
Proof. If A “ tpU ,φ q | α P Au is an atlas, we define a new atlas A˚ of all
α α
charts on M compatible with every chart in A. To be compatible with every
chart in A means that if pU,φq P A˚, φ “φ˝φ´1issmoothforallαP A.
UU α
˚ α
Weshouldjustify that A is an atlas. This involves checking conditions (i)
and(ii) in Definition 4.
Clearly (i) is satisfied, because A˚ contains A and A covers M.
For (ii), we suppose pU,φ q and pV,φ q are elements of A˚. We show that
U V
the homeomorphism φ is smooth. It suffices to show that φ is smooth
VU VU
in a neighborhood of each point φ ppq for φ P A. To that end, consider the
α α
neighborhood φ pU XUXVqofφ ppqinφ pUXVq. Itsufficestoshowthat
U α α U
φ is smooth when restricted to this neighborhood; that is, we want to show
VU
that
φ | : φ pUXVXU qÑφ pUXVXU q
VU φ pUXVXU q U α V α
U α
is smooth. LetW “ UXVXU . φ | canberealizedasthecomposition
α VU φ pWq
U
of two smooth transition functions as follows:
φ | “φ ˝φ´1˝φ ˝φ´1| “pφ ˝φ´1q| ˝pφ ˝φ´1q |
VU φ pWq V α α U φ pWq V α φ pWq α U φ pWq
U U α U
φ |
VU φ pWq
φ pWq U φ pWq
U V
φ | φ |
U U φ pWq VU φ pWq
α U φ pWq α α
α
Sinceeachofφ andφ is smoothbyassumption,thensoistheircompos-
U U VU
α α
ite and so φ is smooth at φ ppq. Therefore, it is smooth.
VU α
Nowfinally, we need to justify that A˚ is maximal. Clearly any atlas con-
taining A must consist of elements of A˚. So A˚ is maximal and unique.
Definition 8. A topological manifold M with a smooth structure is called a
smoothmanifoldofdimensiond. Sometimesweuse Md todenotedimension
d.
Remark 9. We can also talk about Ck manifolds for k ą 0. But this course is
aboutsmoothmanifolds.
Example10.
(i) Rd with the chart consisting of one element, the identity, is a smooth
manifold.
(ii) Sd Ď Rd`1 is clearly a Hausdorff, second-countable topological space.
` ~ d ´ ~ d
Let U “ tx P S | xi ą 0u and let U “ tx P S | xi ă 0u. We have
i i
3
charts φ : U` Ñ Rd and ψ : U´ Ñ Rd given by just forgetting the i-th
i i i i
coordinate. Note that φ ˝φ´1 (and ψ ˝φ´1) are both maps defined by
2 1 2 1
b
py2,...,yd`1q Ñ ´ 1´y2´...´y2 , y3,. . . , yd`1¯.
2 d`1
This is smooth on an appropriate subset of
` ! 2 2 )
φ pU q “ py2,...,y q | y `...`y ă1
1 1 d`1 2 d`1
given by y2 ą 0 (resp. y2 ă 0q. The reason that y2 ą 0 is the appropriate
` ` ~ d
subset is because U XU “txPS |x1ą0andx2ą0u,andwewant
´1 1 2 `
φ py ,...,y q to be in U so that it’s in the domain of φ .
1 2 d`1 2 2
From this it follows that Sd is a smooth manifold. We should be careful
~ d `
to note that each x P S has some xi ‰ 0, so lies in one of the sets U or
´ i
U .
i
(iii) Similarly the real projective space RPd “ Sd{t˘1u identifying antipodal
points is a smooth manifold.
Lecture 2 10 October 2015
Example11. Furtherexamples. Continuedfromlasttime.
2 ~ ~
(iv) Consider the equivalence relation on R given by x „ y if and only if
x1 ´y1 P Z, x2 ´y2 P Z. Let T denote the quotient topological space the
2-dimensional torus. Any unit square Q in R2 with vertices at pa,bq, pa`
1,bq,pa,b`1q,andpa`1,b`1qdeterminesahomeomorphismπ: intQ „ UpQqĂ
T, withUpQq “ πpintQqopeninT. Theinverseisachart. Giventwodif-
ferent unit squares Q ,Q2, we get the coordinate transform φ which is
1 21
locally (but not globally) just given by translation. This gives a smooth
structure on T. Similarly define the n-torus Tn “ Rn{Zn as a smooth
manifold.
Definition12. Let Mm, Nn besmoothmanifoldswithgivensmoothstructures.
Acontinuous map f : M Ñ N is smooth if for each p P M, there are charts
pU,φ q,pV,ψ qwith p P U, fppq P V, such that f “ ψ ˝ f ˝φ´1 is smooth.
U V V U
p f ppq
P P
UXf´1pVq f V
φ ψ
U V
φ pUXf´1pVqq ψ pVq
U f V
Note that since the coordinate transforms for different charts are diffeo-
morphisms,thisimpliesthattheconditionthat f is smoothholdsforallcharts
pU1,φ 1q, pV1,ψ 1q with p P U1, fppq P V1.
U V
4
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