Katharina Neusser
Global Analysis
September 14, 2022
This work is licensed under a Creative Commons Attribution-Share Alike 4.0 Unported
License.
Motivation
Analysis in Rn:
• It is concerned with the study of differential)le/smooth functions
/ : U ->• Rm,       U C Rn open.
• Sometimes already other domains occurred:
— Method of Lagrange multipliers to find local extrema of functions / : R2 —>■ R subject to the condition that (x,y) 6 g_1(0) for g : R2 ->• R.
— Theorems of Gaufi, Green and Stokes: domains called curves and surfaces appear.
Such domains are called submanifolds (with or without boundary) in Rn.
Plan of the course:
• Generalize the differential and integral calculus from open subsets of Rn to submanifolds of Rn, which leads also naturally to the notion of abstract manifolds.
• Manifolds can be equipped with various geometric structures and as such they become objects of modern differential geometry:
— Hypersurfaces in Rn inherit from the inner product in Rn a Rie-mannian metric, -w Riemannian submanifolds of Rn.
— Riemannian manifolds
— Symplectic manifolds
v
Motivate
Groups
appear as symmetry groups of geometric structures appear in the study of PDEs
Contents
Motivation v
1 Smooth Manifolds
2 The Tangent Bundle
vii
Chapter 1
Smooth Manifolds
1.1 Submanifolds of Rn
We want to identify a class of ,nice' subsets of Rn, which will be called submanifolds of Rn, on which we can develop a differential and integral calculus as on open subsets of Rn.
For m < n consider the inclusion
Recalling that differentiability is a local concept, we may consider subsets of Rn that locally have the form of (1.1).
Definition 1.1. A subset M C Rn admits local m-dimensional triviali-sations, if for every x 6 M there exists an open neighbourhood U of x in Rn, an open subset V of Rn and a diffeomorphism (f) : U —>■ V such that
<j>(U n M) = V n Rm C Rm x Rn"m = Rn.
We may also consider graphs of smooth functions g : Rm —> Rn m:
gr(5) := {(x, g(x)) : x e Rm} C Rm X Rn"m = Rn. (1.2) Localising (1.2) yields:
Definition 1.2. A subset M C Rn is locally the m-dimensional graph of a smooth function, if for every x 6 M there exists an open neighbourhood U of x in Rn, an m-dimensional subspace W C Rn, an open subset V C W and a smooth function g : V —>■ W1- such that
UHM = gr(g) C W ©      = Rn,
1
2
Smooth Manifolds
where W1- = {x 6 Rn : (x, w) = 0 6 W} is the orthogonal compliment of W in Rn with respect to the standard inner product (•, •) : I" x Rn -)• 1.
We may also consider zero sets of smooth regular functions. A smooth function
f :U ^ Wn-'m1    U C Rn open ,
is called regular at y 6 U, if the derivative Dyf : Rn —>■ Rn_m is surjective. It is called regular, if / is regular at all points of U. Note that if / is regular at y, then it is so locally around y, since the rank of Dyf is locally constant.
Definition 1.3. A subset M C Rn is locally the m-dimensional zero set of a regular smooth function, if for every x 6 M there exists an open neighbourhood U of x in Rn and smooth function /:[/—>■ Rn_m that is regular at x such that
MnU = f-\0) = {y(EU:f(x)=0}.
Yet another nice class of subsets arise as images of open subsets of Rm under immersions into Rn:
Definition 1.4. A subset M C Rn admits local m-dimensional parametri-sations, if for every x 6 M there exists an open neighbourhood U of x in Rn, an open subset V C Rm and a smooth map    : V —>■ U such that
• Dyip : Rm ->• Rn is injective for all y e V, and
• i\) induces a homeomorphism onto its image:    : V = M CiU = Im^).
Theorem 1.5. Assume M C Rn is a subset ofWn. Then the following are equivalent:
(a) M admits local m-dimensional trivialisations.
(b) M is locally the m-dimensional zero set of a regular smooth function.
(c) M is locally the m-dimensional graph of a smooth function.
(d) M admits local m-dimensional parametrisations.
l.i Submanifolds of W'
3
The proof is based on the Inverse Function Theorem, which we recall now:
Theorem 1.6 (Inverse Function Theorem). Let U C Rn be an open subset, F : U —>Wl a smooth map, and x e U. If the derivative DXF : Rn —>• Rn of F at x is an isomorphism, then there exist open neighbourhoods V of x and W of F(x) such that F(V) = W and
F\V:V -+W
is a diffeomorphism.
Proof. See Analysis/Calculus class. □
An immediate corollary is:
Corollary 1.7 (Implicit Function Theorem). Assume m < n. Suppose
/ : Rm x Rn"m Rn"m
is a smooth function with /(0, 0) = 0 and
d2f(0, 0) := DmF\Rn-m : R""m ^ R""m
is an isomorphism. Then there exists locally a unique solution g(x) of f(x, g(x)) = 0 and x i->- g{x) is smooth.
Proof. Consider F : RmxRn"m RmxRn"m given by F(x, y) = (x, f(x, y)). Note that F is smooth, F(0, 0) = (0, 0) and
D^F-{ * a2/(0,o)
is invertible. By Theorem 1.6, F-1 exists locally around (0, 0) and is smooth. By construction of F, the local inverse F~x is of the form F~x(u,v) = (u,G(u,v)) with G smooth. Hence,
f(x,y)=0 « F(x,y) = (x,0)
(x,y) = F~1(x,0) = (x,G(x,0)) y = G(x,0) =: g(x).
□
Smooth Manifolds
Proof of Theorem 1.5. We start with showing
(b) Assume i£M,[/,l/cl" open and 0 : U —>■ V a diffeommorphism as in Definition 1.1. Set / := vr o 0 : U ->• Rn"m, where vr : Rm x Rn"m Rn_m is the natural projection. By construction, /_1(0) = UdM and / is smooth. Moreover,
Dyf = Dmir o Dy<j> = vr o Dy<j> : Rn = Rn ->• Rn_m
is surjective for all y 6 U.
(c) Assume x 6 M and /:£/—>■ Rn_m as in Definition 1.3. Then AJ : Rn ->• Rn"m is surjective and ker(AJ) =: IV C Rn an Tridimensional subspace. Identify Rn = IV © IV1- and write x = w + wV Then Dxf\w± : II7-1 —>■ Rn_fc is an isomorphism. Hence, by Corollary 1.7, there exists open neighbourhoods V C IV and V C IV1- of w respectively w1- and a smooth function 5» : V —>■ V C IV1- such that
Mn(7xK') = /_1(o) n (V x I/') = O, e V}.
(d) Assume a; € M, [/, V C IV, and 5 : V —>• IV1- as in Definition 1.2. Now consider the map ^ : V —>■ II7 © IV1- = Rn given by ip(v) = (v, g(v)). It is smooth and ip(V) = M n £/. Moreover, since o -0 = Id, where ttw : IV © IV1- —>■ IV is the natural projection, -0 is a homeomorphism onto its image. Also, for Dvip : IV —>■ IV © IV1- one has
D„^(ti;) = (w,Dvgw) = (0,0) w = 0.
(a) Assume a; £ M, V C Rm and U C Rn open and tp : V —>• £/ as in Definition 1.4. Without loss of generality we may assume 0 6 V and ■0(0) = x. Then IV := Im(DO'0) C Rn is an m-dimensional subspace and we identify Rn = IV © W^. Now define
# : V x IV^ ->• Rn #(t>, w) := 0(f) + to.
Note that #(0, 0) = x and with respect to the identification Rn = IV © IV1- the derivative of <P at (0, 0) has the form
/i?o0 0 " I  0 ldw,
l.i Submanifolds of W'
5
Hence, D(0;o)^ : W © W —> Wl is an isomorphism and, by Theorem 1.6, there exist open subsets V\ C V, Vi C W1- and 5 C Rn with x € S such that # : V\ x V2 —>■ <5 is a diffeomorphism.  Since -0 :
V —>■ £/ H M is a homeomorphism, there exists an open subset 5 C Rn with ^(14) = S n M. Set [7 := C/ n 5 n 5 C Mn, which is an open neighbourhood of x by construction, and define
</>:= ($-%:U^<l>(U) :=V.
Then 0 is a diffeomorphism between the open subsets U C Rn and
V C Vi x V2 C V xW1- C Rn. Moreover, if y e M n U, then in particular y 6 M (~l 5, which implies that there exists v\ 6 Vi such that ip(vi) = y. Since y 6 S, this shows 0(y) = (t>i,0). Conversely, if («1, 0) eV n IV, then <P(vu 0) = € U (~1 M by definition of ^. Hence,       D M) = V n IV.
□
Definition 1.8. Assume 1 < m < n are integers. A subset M C Rn is called a (smooth) submanifold of Rn of dimension m, if M satisfies any of the equivalent conditions in Theorem 1.5.
Note that as a subset of Rn a submanifold M C Rn inherits a topology from Rn, namely the subspace topology:
f/cMis open   •<=>- U = U H M for some open subset [7 C Rn.
Remark 1.9.
• If one replaces smooth/C00 everywhere by C for 1 < r < 00 or by C", one obtains the notion of C-submanifolds respectively real analytic submanifolds of Rn.
• Similarly, if one replaces 1 by C and smooth by holomorphic, one obtains complex submanifolds of Cn.
• Replacing C°° in Definition 1.1 by C° leads to topological submanifolds of Rn. In this case, not all the definitions 1.1-1.4 are equivalent!
Some trivial examples and natural constructions:
6
Smooth Manifolds
Example 1.1 (Open subsets). Any open subset U C Rn is an n-dimensional submanifold of Rn and all n-dimensional submanifolds of Rn are of this form. More generally, any open subset of a submanifold in Rn is again a submanifold (of the same dimension). Note also that of course any open subset of Rn can be seen as an n-dimensional submanifold of Rd via the standard inclusion Rn     Rd for n < d.
Example 1.2 (Products). If M C Rn and K C R^ are submanifolds of dimensions m respectively k of Rn respectively R^, then
M x K c Rn x Re = Rn+e
is an m + k dimensional submanifold of Rn x R^.
Some non-trivial examples:
Example 1.3. Consider Rm+1 equipped with its standard inner product (•, •) : Rm+1 x Rm+1 —>• R. Then the m-dimensional (unit) sphere
Srn := {x e Rm : ||^|| = 1} C Rm+1
is the prototypical example of an m-dimensional submanifold of Rm+1. For m = 1, one gets the unit circle S1 in R2. To see this, note that Sm can be described globally as the zero set of the smooth function / : Rm+1 \ {0} —>■ R given by f(x) = (x,x) - 1, i.e. /_1(0) = Sm. Since for any x e Rm+1 \ {0} and v 6 Rm+1 one has
d d Dxfv = —\t=0(x + tv,x + tv) - 1 = —\t=0(x,x) +2t(x,v) +t2(v,v)
= 2{x,v),
the derivative Dxf : Rm+1 —>■ R is surjective by non-degeneracy of (•,•). Hence, / is regular.
Example 1.4. For fixed positive real integers ai, ...,am+i 6 R>o consider the function
/ : Rm+1 \ {0} R
k   x2        m+1 x2
f(x1,...,xm+1):=J2-^-        ~k - 1-
i=l Ui       i=k+l Ui
It is smooth and regular. Hence, /_1(0) := M is an m-dimensional submanifold of Rm+1. Depending on k, these submanifolds are m-dimensional ellipsoids or hyperboloids.
l.i Submanifolds of W'
7
Example 1.5. Consider Cm = R2m as real vector space. Then
Tm := {z e Cm : \Zl \ = ... = \zm\ = 1} C R2m
is an m-dimensional submanifold of R2m, since /_1(0) = Tm, where / : Cm \ {0} —>■ R'm is the smooth regular function given by
f(zu...,zm) = (1^1 - l,...,|zm - 1).
Of course, also
Tm 2á S1 x ... x S1 c K2 x ... xl2 = R2m,
V-v-' V-v-'
m—times ra—times
so Tm is an m-dimensional submanifold of R2m by Examples 1.3 and 1.2. It is called the m-dimensional torus.
Example 1.6. Consider the vector space Hom(Rn, Rn) of linear maps from Rn to Rn. Via a choice of basis of Rn,
Hom(Rn,Rn) ^ Mnxn(R) = R™2,
where M„xn(R) denotes the vector space of real n x n matrices. Since the determinant det : M„xn(R) —>■ R is continuous (polynomial in the eneries of the matrix), the subset
GL(n,R) := {A e Mnxn(R) : det (A) + 0} C Mnxn(R)
is open and as such an n2-dimensional submanifold of M„xn(R) = R™2. Note that GL(n, R) is also a group with respect to matrix multiplication. It is called the general linear group.
In fact, det : GL(n,R) :—>■ R is smooth and also regular, since for any A e GL(n, R) one has
(DA det) (A) = ^|t=0 det (A + tA)
= l|t=0det((l + i)A) = ^|i=0(l + t)ndet(A) = ndet(A) ^ 0,
which shows that D^det : GL(n, R) —>■ R is surjective for all A 6 GL(n, R). Hence, also / := det—1 : GL(n, R) :—>■ R is a smooth regular function. Therefore,
SL(n,R) := /_1(0) = {A (E GL(n,R) : det A = 1} C Mnxn(R)
8
Smooth Manifolds
is an (n2 — l)-dimensional submanifold of M„xn(R). It is also a group with respect to matrix multiplication, called the special linear group.
Now consider the map
/ : GL(n, R) ->• Mnxn(R) f(A) := AAl - Id
and set
0(n) := /_1(0) = {A <E GL(n,R) : AA* = Id}.
Note that /(A)1 = f(A). Hence, / has values in the subspace M^X™(R) C Mnxn(M) of symmetric n x n-matrices. The function
/ : GL(n,R)     M^R) ^ R^
is obviously smooth. To see that it is also regular, note that (A, B) i->- ABl is bilinear as a map Mnxn(M) x Mnxn(M) —>■ M„xn(R). Therefore, for any A e GL(n,R) and B € Mnxn(R), one has DAfB = ABl + BAK So, if A e 0(n) and 5 e M*yx™(R) is arbitrary, then for 5 :=        one has
DAfB = ^(AA^S1 + SAA^) = 1(5' + 5) = S,
=Id =Id
which shows that DAf : Mnxn(R) ->• M*yx™(R) ^ R11^ is surjective for any A 6 0(n). Therefore, the set 0(n) of orthogonal n x n-matrices is a submanifold of R™2 of dimension n(n~1). It is also closed under matrix multiplication and hence a group, called the orthogonal group.
For submanifolds of Rn, we have an obvious notion of defining smooth maps between them:
Definition 1.10. Suppose M C Rn is an m-dimensional submanifold.
• A map / : M —>■ M.e is smooth, if for every point x 6 M there exists an open neighbourhood U of x in Rn and a smooth function / : U —>■ M.e such that f\Mn~ = f\Mn~.
• For a fc-dimensional submanifold K C R^ a map f : M K is smooth, if it is is smooth as a map M —>■ M.e.