Analysis On Manifolds
1 Manifolds
1.1 Topological manifolds
Definition Let X be some set and T a set of subsets of X. A pair (X, T ) is a topological space (with
O ∈ T being called open) if
1. X ∈ T , ∅ ∈ T .
2. Arbitrary unions of families of open subsets are open.
3. The intersection of finitely many open subsets is open.
Definition A map f : (X, T ) → (Y, U) between two topological spaces is called
• Continuous if U ∈ U implies that f −1 (U ) ∈ T , that is, the pre-image of open is open.
• Homeomorphic if it is bijective and continuous with continuous inverse.
Definition A topological space (X, T ) is Hausdorff if for every pair x ̸= y of points in X, there exists
Ux , Uy ∈ T such that x ∈ Ux , y ∈ Uy and Ux ∩ Uy = ∅.
Any metric space with the metric topology is Hausdorff.
Definition A topological space (X, T ) is second countable if there exists a countable set B ⊂ T such
that any open set can be written as a union of sets in B. In such case, B is called a (countable) basis for
the topology T .
Defintion A topological space M is a topological manifold of dimension n (or a topological n-manifold)
if
1. M is a Hausdorff space;
2. M is second countable;
3. M is locally euclidean of dimension n, that is, for any point p ∈ M there exists an open subset
U ⊂ M with p ∈ U , and open subset V ⊂ Rn and a homeomorphism ϕ : U → V .
Notation: we call (U, ϕ) a coordinate chart of a coordinate neighborhood U and an associated coordinate
map ϕ : U → V onto V = ϕ(U ) ⊂ Rn . We say a chart is centered at p ∈ U if ϕ(p) = 0.
1.2 Differentiable manifolds
Definition A map f : U → V between open sets U ⊂ Rn and V ⊂ Rm is in C r (U, V ) or of class C r , if
it is continioulsy differentiable r times. It is called a C r diffeomorphism if it is bijective and of class C r
with inverse of class C r . We say that f is smooth, or of class C ∞ if it is of class C r for every r ≥ 1.
Theorem (chain rule) Let U ⊂ Rn and V ⊂ Rk be open sets and f : U → Rk , g : V → Rm two
continuously differentiable functions such that f (U ) ⊂ V . Then,
• The function g ◦ f : U → Rm is continuously differentiable and its total derivative at a point x ∈ U
is given by
D(g ◦ f )(x) = (Dg)(f (x)) ◦ Df (x).
• Denote x = (x1 , . . . , xn ) and y = (y 1 , . . . , y k ) the coordinates on the respective euclidean spaces
and f = (f 1 , . . . , f k ) and g = (g 1 , . . . , g m ) the components of the functions. Then the partial
derivatives are given by
k
∂g i ◦ f X ∂g i ∂f r ∂g i ∂f r
(x) = (f (x)) (x) = (f (x)) (x), 1 ≤ i ≤ m, 1 ≤ j ≤ n
∂xj r=1
∂y r ∂xj ∂y r ∂xj
1
,Definition We say that two charts (U1 , ϕ1 ) and (U2 , ϕ2 ) on a topologival manifold M are compatible
if either U1 ∩ U2 = ∅ or if the transition map ϕ1 ◦ ϕ−1 2 : ϕ2 (U1 ∩ U2 ) → ϕ1 (U1 ∩ U2 ) is a smooth
diffeomorphism.
Definition A smooth atlas Sis a collection A = {ϕa : Ua → Va | a ∈ A} of pairwise compatible charts
that cover M (that is, M = a∈A Ua ). Two smooth atlases are equivalent if their union is also a smooth
atlas.
Definition A differentiable structure, or more precisely a smooth structure, on a topological manifold
is an equivalence class of smooth atlases.
Definition A smooth manifold of dimension n is a pair (M, A) of a topological n-manifold M and a
smooth structure A on M ( a differentiable manifold is just a space covered by charts with differentiable
transition maps).
We denote ri : Rn → R the standard coordinates on Rn . If ej denotes the jth standard basis vector in Rn ,
then ri (ej ) = δji . If (U, ϕ) is a chart of a manifold, then xi = ri ◦ϕ will denote the i-th component of ϕ and
denote ϕ = (x1 , . . . , xn ) and, when convenient, (U, ϕ) = (U, x1 , . . . , xn ). For p ∈ U , (x1 (p), . . . , xn (p)) is
a point in Rn . The functions x1 , . . . , xn are called (local) coordinates on U .
Given two manifolds (M1 , A1 ) and (M2 , A2 ) we can define the product manifold M1 × M2 using the
product topology and the atlas {(U1 × U2 , (ϕ1 , ϕ2 )) | (Ui , ϕi ) ∈ Ai }
Lemma (smooth manifold lemma) Let M be a set. Assume we are given a collection {Ua | a ∈ A}
of subsets of M together with bijections ϕa : Ua → ϕa (Ua ) ⊂ Rn , where ϕa (Ua ) is an open subset of Rn .
Assume that also the following hold:
• For each a, b ∈ A, the sets ϕa (Ua ∩ Ub ) and ϕb (Ua ∩ Ub ) are open in Rn ;
• If Ua ∩ Ub ̸= ∅, the map ϕb ◦ ϕ−1
a : ϕa (Ua ∩ Ub ) → ϕb (Ua ∩ Ub ) is smooth;
• Countably many of the sets Ua cover M ;
• If p ̸= q are points in M , either there exists a such that p, q ∈ Ua or there exists a, b with Ua ∩Ub = ∅
such that p ∈ Ua and q ∈ Ub .
Then M has a unique smooth manifold structure such that each (Ua , ϕa ) is a smooth chart.
1.2.1 Quotient manifolds
Proposition Assume F : X → Y is a map between topological spaces and ∼ is an equivalence reation
on X. Let F be constant on each equivalence class [p] ∈ X/ ∼, and denote F̃ : X/ ∼→ Y , F̃ ([p]) := F (p)
for p ∈ X, the map induced by F on the quotient. Then, F̃ is continuous if and only if F is continuous.
Theorem If M is a topological space and ∼ an equivalence relation such that the projection map
π : M → M/ ∼ is open, then:
• π maps a basis for the topology of M into a basis for the topology M/ ∼, thus if M is second
countable, then M/ ∼ is second countable;
• The quotient space M/ ∼ is Hausdorff if and only if R := {(x, y) ∈ M × M | x ∼ y} (i.e. the graph
of ∼) is closed in M × M .
1.3 Smooth maps and differentiability
Defintion A function f : M → R from a smooth manifold M of dimension n to R is smooth, or of class
C ∞ , if for any smooth chart (ϕ, V ) for M the map f ◦ ϕ−1 : ϕ(V ) → R is smooth as a euclidean function
on the open subset ϕ(V ). We denote the space of smooth functions by C ∞ (M ).
Proposition Let M be a smooth n-manifold and f : M → R a real-valued function on M . Then the
following are equivalent:
• f ∈ C ∞ (M );
2
, • M has an atlas A such that for every chart (U, ϕ) ∈ A, f ◦ ϕ−1 is C ∞ ;
• for every point p ∈ M , there exists a smooth chart (V, ψ) for M such that p ∈ V and the function
f ◦ ψ −1 is C ∞ on the open subset ψ(V ).
Definition Let F : M1 → M2 be a continuous map between two smooth manifolds of dimension n1 and
n2 respectively. We say that F is smooth (or of class C ∞ ) if for any chart (ϕ1 , V1 ) of M1 and (ϕ2 , V2 ) of
M2 , the map ϕ2 ◦ F ◦ ϕ−1 ∞
1 is smooth as a euclidean function. We denote by C (M1 , M2 ) all functions
∞ −1
F : M1 → M2 of class C . The map F̃ := ϕ2 ◦ F ◦ ϕ1 is called the coordinate representation of F .
Proposition Let M be a smooth manifold of dimension n. Then F : M → Rm is smooth if and only if
for all smooth charts (U, ϕ) of M , the function F ◦ ϕ−1 is smooth.
Proposition Let M be a smooth manifold of dimension n. Then F : M → Rm is smooth if and only if
for all smooth charts (U, ϕ) of M , the function ϕ ◦ F is smooth.
Proposition Let M, N, P be three smooth manifolds, and suppose that F : M → N and G : N → P
are smooth. Then G ◦ F ∈ C ∞ (M, P ).
Proposition (Smoothness is a local property) Let F : M → N be a continuous function and let
{Ui }i∈I be an open cover for M . Then F |Ui : Ui → N is smooth for every i ∈ I if and only if F : M → N
is smooth.
Proposition (Glueing lemma for smooth maps) Let M and N be two smooth manifolds and let
{Ua : a ∈ A} be an open cover of M . Suppose that for each a ∈ A we are given a smooth map
Fa : Ua → N such taht the maps agree on overlaps: Fa |Ua ∩Ub = Fb |Ua ∩Ub for all a, b ∈ A. Then there
exists a unique smooth map F : M → N such that F |Ua = Fa for each a ∈ A.
Definition A diffeomorphism F between two smooth manifolds M1 and M2 is a bijective map such that
F ∈ C ∞ (M1 , M2 ) and F −1 ∈ C ∞ (M2 , M1 ). Two smooth manifolds M1 and M2 are called diffeomorphic
if there exists a diffeomorphism F : M1 → M2 between them.
Definition A map F : M1 → M2 is a local diffeomorphism if every point p ∈ M1 has a neighbourhood
U such that F (U ) is open in M2 and F |U : U → F (U ) is a diffeomorphism.
1.4 Manifolds with boundary
Define
Hn = {x ∈ Rn : xn ≥ 0}, ∂Hn = {x ∈ Rn : xn = 0}
Definition A topological space M is a topological manifold with boundary of dimension n, or topological
n-manifold with boundary, if it has the following properties:
• M is a Hausdorff space
• M is second countable
• M is locally homeomorphic to Hn , any point x ∈ M has a neighbourhood that is homeomorphic
to a (relatively) open subset of Hn .
A chart on M is a pair (U, ϕ) consisting of an open set U ⊂ M and a homeomorphism ϕ : U → ϕ(U ) ⊂ Hn .
Definition Let U ⊂ Hn be a relatively open set. A map f : U → Rm is r times continuously differentiable
(of class C r ) if there exists an open set Ũ ⊂ Rn and a map f˜ ∈ C r (Ũ , Rm ) such that U ⊂ Ũ and f˜|U = f .
The function f is said to be smooth (class C ∞ ) if f is C r for every r ≥ 1.
Definition A smooth manifold with boundary of dimension n is a pair (M, A) of a topological n-manifold
with boundary M and a smooth differentiable structure A = {(Ua , ϕa ) | a ∈ A} on M . The boundary of
M is defined as [
∂M := ϕ−1 n
a (ϕa (Va ∩ ∂H )
a∈A
Proposition The boundary is well-defined.
Proposition Any diffeomorphism between smooth n-manifolds F : M → N satisfies F (∂M ) = ∂N .
3
1 Manifolds
1.1 Topological manifolds
Definition Let X be some set and T a set of subsets of X. A pair (X, T ) is a topological space (with
O ∈ T being called open) if
1. X ∈ T , ∅ ∈ T .
2. Arbitrary unions of families of open subsets are open.
3. The intersection of finitely many open subsets is open.
Definition A map f : (X, T ) → (Y, U) between two topological spaces is called
• Continuous if U ∈ U implies that f −1 (U ) ∈ T , that is, the pre-image of open is open.
• Homeomorphic if it is bijective and continuous with continuous inverse.
Definition A topological space (X, T ) is Hausdorff if for every pair x ̸= y of points in X, there exists
Ux , Uy ∈ T such that x ∈ Ux , y ∈ Uy and Ux ∩ Uy = ∅.
Any metric space with the metric topology is Hausdorff.
Definition A topological space (X, T ) is second countable if there exists a countable set B ⊂ T such
that any open set can be written as a union of sets in B. In such case, B is called a (countable) basis for
the topology T .
Defintion A topological space M is a topological manifold of dimension n (or a topological n-manifold)
if
1. M is a Hausdorff space;
2. M is second countable;
3. M is locally euclidean of dimension n, that is, for any point p ∈ M there exists an open subset
U ⊂ M with p ∈ U , and open subset V ⊂ Rn and a homeomorphism ϕ : U → V .
Notation: we call (U, ϕ) a coordinate chart of a coordinate neighborhood U and an associated coordinate
map ϕ : U → V onto V = ϕ(U ) ⊂ Rn . We say a chart is centered at p ∈ U if ϕ(p) = 0.
1.2 Differentiable manifolds
Definition A map f : U → V between open sets U ⊂ Rn and V ⊂ Rm is in C r (U, V ) or of class C r , if
it is continioulsy differentiable r times. It is called a C r diffeomorphism if it is bijective and of class C r
with inverse of class C r . We say that f is smooth, or of class C ∞ if it is of class C r for every r ≥ 1.
Theorem (chain rule) Let U ⊂ Rn and V ⊂ Rk be open sets and f : U → Rk , g : V → Rm two
continuously differentiable functions such that f (U ) ⊂ V . Then,
• The function g ◦ f : U → Rm is continuously differentiable and its total derivative at a point x ∈ U
is given by
D(g ◦ f )(x) = (Dg)(f (x)) ◦ Df (x).
• Denote x = (x1 , . . . , xn ) and y = (y 1 , . . . , y k ) the coordinates on the respective euclidean spaces
and f = (f 1 , . . . , f k ) and g = (g 1 , . . . , g m ) the components of the functions. Then the partial
derivatives are given by
k
∂g i ◦ f X ∂g i ∂f r ∂g i ∂f r
(x) = (f (x)) (x) = (f (x)) (x), 1 ≤ i ≤ m, 1 ≤ j ≤ n
∂xj r=1
∂y r ∂xj ∂y r ∂xj
1
,Definition We say that two charts (U1 , ϕ1 ) and (U2 , ϕ2 ) on a topologival manifold M are compatible
if either U1 ∩ U2 = ∅ or if the transition map ϕ1 ◦ ϕ−1 2 : ϕ2 (U1 ∩ U2 ) → ϕ1 (U1 ∩ U2 ) is a smooth
diffeomorphism.
Definition A smooth atlas Sis a collection A = {ϕa : Ua → Va | a ∈ A} of pairwise compatible charts
that cover M (that is, M = a∈A Ua ). Two smooth atlases are equivalent if their union is also a smooth
atlas.
Definition A differentiable structure, or more precisely a smooth structure, on a topological manifold
is an equivalence class of smooth atlases.
Definition A smooth manifold of dimension n is a pair (M, A) of a topological n-manifold M and a
smooth structure A on M ( a differentiable manifold is just a space covered by charts with differentiable
transition maps).
We denote ri : Rn → R the standard coordinates on Rn . If ej denotes the jth standard basis vector in Rn ,
then ri (ej ) = δji . If (U, ϕ) is a chart of a manifold, then xi = ri ◦ϕ will denote the i-th component of ϕ and
denote ϕ = (x1 , . . . , xn ) and, when convenient, (U, ϕ) = (U, x1 , . . . , xn ). For p ∈ U , (x1 (p), . . . , xn (p)) is
a point in Rn . The functions x1 , . . . , xn are called (local) coordinates on U .
Given two manifolds (M1 , A1 ) and (M2 , A2 ) we can define the product manifold M1 × M2 using the
product topology and the atlas {(U1 × U2 , (ϕ1 , ϕ2 )) | (Ui , ϕi ) ∈ Ai }
Lemma (smooth manifold lemma) Let M be a set. Assume we are given a collection {Ua | a ∈ A}
of subsets of M together with bijections ϕa : Ua → ϕa (Ua ) ⊂ Rn , where ϕa (Ua ) is an open subset of Rn .
Assume that also the following hold:
• For each a, b ∈ A, the sets ϕa (Ua ∩ Ub ) and ϕb (Ua ∩ Ub ) are open in Rn ;
• If Ua ∩ Ub ̸= ∅, the map ϕb ◦ ϕ−1
a : ϕa (Ua ∩ Ub ) → ϕb (Ua ∩ Ub ) is smooth;
• Countably many of the sets Ua cover M ;
• If p ̸= q are points in M , either there exists a such that p, q ∈ Ua or there exists a, b with Ua ∩Ub = ∅
such that p ∈ Ua and q ∈ Ub .
Then M has a unique smooth manifold structure such that each (Ua , ϕa ) is a smooth chart.
1.2.1 Quotient manifolds
Proposition Assume F : X → Y is a map between topological spaces and ∼ is an equivalence reation
on X. Let F be constant on each equivalence class [p] ∈ X/ ∼, and denote F̃ : X/ ∼→ Y , F̃ ([p]) := F (p)
for p ∈ X, the map induced by F on the quotient. Then, F̃ is continuous if and only if F is continuous.
Theorem If M is a topological space and ∼ an equivalence relation such that the projection map
π : M → M/ ∼ is open, then:
• π maps a basis for the topology of M into a basis for the topology M/ ∼, thus if M is second
countable, then M/ ∼ is second countable;
• The quotient space M/ ∼ is Hausdorff if and only if R := {(x, y) ∈ M × M | x ∼ y} (i.e. the graph
of ∼) is closed in M × M .
1.3 Smooth maps and differentiability
Defintion A function f : M → R from a smooth manifold M of dimension n to R is smooth, or of class
C ∞ , if for any smooth chart (ϕ, V ) for M the map f ◦ ϕ−1 : ϕ(V ) → R is smooth as a euclidean function
on the open subset ϕ(V ). We denote the space of smooth functions by C ∞ (M ).
Proposition Let M be a smooth n-manifold and f : M → R a real-valued function on M . Then the
following are equivalent:
• f ∈ C ∞ (M );
2
, • M has an atlas A such that for every chart (U, ϕ) ∈ A, f ◦ ϕ−1 is C ∞ ;
• for every point p ∈ M , there exists a smooth chart (V, ψ) for M such that p ∈ V and the function
f ◦ ψ −1 is C ∞ on the open subset ψ(V ).
Definition Let F : M1 → M2 be a continuous map between two smooth manifolds of dimension n1 and
n2 respectively. We say that F is smooth (or of class C ∞ ) if for any chart (ϕ1 , V1 ) of M1 and (ϕ2 , V2 ) of
M2 , the map ϕ2 ◦ F ◦ ϕ−1 ∞
1 is smooth as a euclidean function. We denote by C (M1 , M2 ) all functions
∞ −1
F : M1 → M2 of class C . The map F̃ := ϕ2 ◦ F ◦ ϕ1 is called the coordinate representation of F .
Proposition Let M be a smooth manifold of dimension n. Then F : M → Rm is smooth if and only if
for all smooth charts (U, ϕ) of M , the function F ◦ ϕ−1 is smooth.
Proposition Let M be a smooth manifold of dimension n. Then F : M → Rm is smooth if and only if
for all smooth charts (U, ϕ) of M , the function ϕ ◦ F is smooth.
Proposition Let M, N, P be three smooth manifolds, and suppose that F : M → N and G : N → P
are smooth. Then G ◦ F ∈ C ∞ (M, P ).
Proposition (Smoothness is a local property) Let F : M → N be a continuous function and let
{Ui }i∈I be an open cover for M . Then F |Ui : Ui → N is smooth for every i ∈ I if and only if F : M → N
is smooth.
Proposition (Glueing lemma for smooth maps) Let M and N be two smooth manifolds and let
{Ua : a ∈ A} be an open cover of M . Suppose that for each a ∈ A we are given a smooth map
Fa : Ua → N such taht the maps agree on overlaps: Fa |Ua ∩Ub = Fb |Ua ∩Ub for all a, b ∈ A. Then there
exists a unique smooth map F : M → N such that F |Ua = Fa for each a ∈ A.
Definition A diffeomorphism F between two smooth manifolds M1 and M2 is a bijective map such that
F ∈ C ∞ (M1 , M2 ) and F −1 ∈ C ∞ (M2 , M1 ). Two smooth manifolds M1 and M2 are called diffeomorphic
if there exists a diffeomorphism F : M1 → M2 between them.
Definition A map F : M1 → M2 is a local diffeomorphism if every point p ∈ M1 has a neighbourhood
U such that F (U ) is open in M2 and F |U : U → F (U ) is a diffeomorphism.
1.4 Manifolds with boundary
Define
Hn = {x ∈ Rn : xn ≥ 0}, ∂Hn = {x ∈ Rn : xn = 0}
Definition A topological space M is a topological manifold with boundary of dimension n, or topological
n-manifold with boundary, if it has the following properties:
• M is a Hausdorff space
• M is second countable
• M is locally homeomorphic to Hn , any point x ∈ M has a neighbourhood that is homeomorphic
to a (relatively) open subset of Hn .
A chart on M is a pair (U, ϕ) consisting of an open set U ⊂ M and a homeomorphism ϕ : U → ϕ(U ) ⊂ Hn .
Definition Let U ⊂ Hn be a relatively open set. A map f : U → Rm is r times continuously differentiable
(of class C r ) if there exists an open set Ũ ⊂ Rn and a map f˜ ∈ C r (Ũ , Rm ) such that U ⊂ Ũ and f˜|U = f .
The function f is said to be smooth (class C ∞ ) if f is C r for every r ≥ 1.
Definition A smooth manifold with boundary of dimension n is a pair (M, A) of a topological n-manifold
with boundary M and a smooth differentiable structure A = {(Ua , ϕa ) | a ∈ A} on M . The boundary of
M is defined as [
∂M := ϕ−1 n
a (ϕa (Va ∩ ∂H )
a∈A
Proposition The boundary is well-defined.
Proposition Any diffeomorphism between smooth n-manifolds F : M → N satisfies F (∂M ) = ∂N .
3
