MATH 5061 Lecture 1 Jan 13

Assessment Biweekly Hw 50 Take home Final 50

TA Garoming WANG

Please refer to course webpage for more details 1announcements

Q what is Riemannian Geometrygeometry

Undergrad Riemannian topologycontinuation GeometryDiff Geom 7

MATH4030 higher dim MATH5062 calculusanalysisPDEs

intrinsiccurves surfaces geometry Mn gin IR or 1133

j lcurvatures abstract Riemannian

smoochmetric

manifold

we begin with the theory of smooth manifolds

Ref F Warner Foundations of Differential Manifoldsand Lie Groups

J Lee Introduction to Smooth Manifolds

Submanifolds of IRN

Def1 M E B 1k is an n din't submanifold of Bntk

of class CP if V X E M 7 nbd U E B andu

a CP map f U Bk s 1 ifMEIR8

if pki

i dfqa.IR Bk is onto it qe U i.e f is a submersion

Cii Un M f o

Idea MT regular zero set of a CPcreator

valued function

Here k code'm M n dim M

When K 1 we say M is a hypersurface

Examples of submfd inCRN

a sphere 8 f exo Xn c IR I xft c Xi 1

ffix y z X'tg't z 1

f B IR CA and f o 83

dfg 2x 2g ZZ to VofE 8

b Hyperboloid HE exo xn c R Xi Xi in c

is a CA submfd in IR when Otc E IR

C oh 2

HE c o

c

Ic Torus T f z Zn e E IZ l t t 17h12 L

is a CA submanifold of Budin n codim n

d SO n f A E MniR ATA I det A 1

is a co submfd of Muir IR dim 50Cn ncn

why GLI R f A C Mul I det A o E Muiropen

Define f GL'TGR Sym n f At A E Muir EIRlinearsubsp

f A ATA I

Note f o So n

At any A C SOCn dfa B A43 t BTA

is an onto map to Sym n

Given Se Symons dfa Azs Atlas SI A S

prop 1 FAE

i M E Bk is a CP submanifold

iz t X E M I nbd c U e Brith EVE Ruth

and a CP diffeomorphism f n U V

St fl Un M V n Rnx lol

differ Vy u ef t i f r I

t Mru iRnxGo

mi e i

f g Pf by ImplicitFunction Theorem

0e 2 IRW

Iii V X E M 7 nbd EE U EBht and o E W E B

and a CR map g w RuthSt G is a homeomorphism onto its image gCW M n U

wth dojo is 1 1 local parametrization chart

Remark For any two such chartsGi Wi IR in 1,2 as in Ciii

then the transition maps 95 o G is a CP diffeomorphism

Ui Ue mean

gze WERW E R r

I n ii j o g lc 92 1

i i

CP differ 7

Abstract Manifoldslocally n

Idea n manifolds I opensubsets of IR

described by compatible charts into IR

ASSUME n M Hausdorff paracompact topological space

partition of unity

Def't n A CP Atlas on M is a collection of charts Ui dillieS t i Ui c I forms an open cover of M

Iii Xi Ui WithIR are homeomorphisms Vi c I

and the transition maps

log o 05 Qi lui n Uj Ij Ui nu are CP differ

Ui U M

di

wiseY w Rn I

a y 4Joff l ir i l

i i

CP differ 7

Def flui.fi c I Vj Yj jej if their union is an atlas

Def't r An equivalence class of CP atlas on M is called a

differentiable structure of class CP on M

A differential manifold consists of a Hausdorffparacompact topologicalspace M together with an

atlas f Ui fistic I

Remark M connected dim M n well defined

by invariance of domain

ASSUME Mn connected smooth line manifold

nDef N E M is a submanifold if V P E N 7 chart U ol

of M St PE U and 01 N n U E IR is a submfd

Examples of abstract manifolds

a submfds of Rntk

b 7 Co structure on a square not inherited from IR

HIIc S E Bnt can be covered by only two charts

N Xi Xnp NIR qp Xo xn 91Xo Xn

by 924091407 gacxo Xu c Pts

I 1XoX isS gzogicy z differ on B fol

an o

d Real Pwjective Space BP I se f pn8 E's app

Rip IR lo f 1 coverX XX x

2 10

A description of co structure on BPXz

OI X Io IR OI Hi xz722 1 Ez Xz to IR Ezix x Ix

Nee Ei Aki XX Ei Xi k Ato

24 1 us Foi are well defined on IRR

and they form a chart on CRP

EI check this

e Replace IR by E 7 ComplexProjectiveSpace EPdim 2n

Def M is orientable if 7 atlas ICUicolillic I set

all transition maps are orientation preservingi.e det dlOjodi's so

Examples 5h is orientable BUT BIP is NOT when n is even

Smooth Maps between manifolds

Let Mm N be smooch manifolds

DEI A Cts map f M N is smooth

if H X E M I charts Uil for X E M

and Chart V X for Cx E N

S t X fo 01 i flu HCV is smooth

x foxn f a

i e M z U V E N

HAHMB 29th TCHE B

Tofool1

Example M E pith submfd and F IR IR smooth

F M M IR is a smooth snap between manifolds

Def'd A smooth map f M N is called an immersion at p c MPa fash

if 7 charts CU 07 for M V 4 for N s t

d 4 fool is 1 1 at locx

Remark submersion local differ if it is onto bijective

Def f e M N diffeomorphism if f is bijective andboth f f are smooth

Exercise Ip Etta S2