Lecture Notes
Differential Geometry
October 15, 2022
Contents
Contents I
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Notations and Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
3 Quick review: Basic Algebraic Structures . . . . . . . . . . . . . . . . . . . 2
4 Quick review: Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
5 Quick review: Equivalence Classes, Quotient Spaces . . . . . . . . . . . . . 4
6 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
7 Multilinear Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
8 Exterior Product and Generalisation . . . . . . . . . . . . . . . . . . . . . . 19
9 Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
10 Integration of Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . 32
11 Tangent Bundles, Flows of Vector Fields . . . . . . . . . . . . . . . . . . . . 33
12 Lie Derivative of a Vector Field . . . . . . . . . . . . . . . . . . . . . . . . . 37
13 Stokes’ Theorem on Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
14 Closed and Exact k-forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
15 De Rham Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
16 Submanifolds of Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
17 Integration on Compact Manifolds . . . . . . . . . . . . . . . . . . . . . . . 65
18 Abstract Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
19 Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
20 Compactly Supported Cohomology . . . . . . . . . . . . . . . . . . . . . . . 81
21 Poincaré Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
22 Riemannian Metric and the Hodge-? Operator . . . . . . . . . . . . . . . . . 94
Bibliography 102
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
I
,1. INTRODUCTION 1
1 Introduction
From the Lee book [Lee]: "The central idea of calculus is linear approximation". A function
of one variable can be approximated by its tangent line, a curve by a tangent vector (i.e.
velocity vector), a surface in R3 can be approximated by its tangent plane, and a map from
Rn to Rm by its total derivative. Here it comes the importance of tangent spaces.
Main idea: in order to study tangent vectors, we identify them with "directional deriva-
tives". In particular, there is a natural one-to-one correspondence between geometric tan-
gent vectors and linear maps from C ∞ (Rn ) to R satisfying the product rule. Such maps
are called derivations.
Remark 1.1. Points or vectors? We can think of elements of Rn either as points
or vectors. As points, their only property is their location, given by the coordinates
(x1 , . . . , xn ) on a chosen basis. As vectors, they are characterized by a direction
and a magnitude, but their location is irrelevant (translational invariance). So given
v ∈ Rn , v = i v i ei = v i ei , it can be seen as an arrow with its initial point anywhere
P
in Rn . So, if we think about a vector tangent to the border of the sphere at a point
a, we imagine the vector as living in a copy of Rn with its origin translated to a.
2 Notations and Conventions
In these notes the Einstein summation convention will be used. It means that the sum
symbol will be omitted when it is clear with respect to which index we are summing. For
instance we will write
v i ei
instead of
X
v i ei .
i
Given a map f from a set X to a set Y , we will denote it by
f: X →Y .
We will use the arrow "7→" to denote how each element of X is mapped into Y through f .
For instance, given x1 , x2 ∈ X, y ∈ Y , we will either use the notation:
x1 7→ f (x1 )
or
x2 7→ y .
We will use the symbol ≡ in order to define mathematical objects. We will use the latin
abbreviation "i.e." (id est) to explain concepts in a more detailed way, and the abbreviation
"e.g." (exempli gratia) to give examples.
,3. QUICK REVIEW: BASIC ALGEBRAIC STRUCTURES 2
3 Quick review: Basic Algebraic Structures
Definition 3.1 (Operation). Let G be a set. The map:
· : G × G −→ G
(a, b) 7−→ a · b
is called a (binary) operation on G. Sometimes such a map is also denoted by the symbol
+.
Definition 3.2 (Group). Given a set G and an operation · on such set, we will call such set
with the operation (i.e. the couple (G, ·)) a group if the following properties are satisfied
for all a, b, c ∈ G:
• a · b ∈ G, (closure property)
• a · (b · c) = (a · b) · c, (associativity)
• ∃ 1 ∈ G such that 1 · g = g, ∀ g ∈ G, (existence of the identity element)
• ∀ g ∈ G, ∃ g −1 ∈ G so that g · g −1 = g −1 · g = 1. (existence of the inverse element)
For the sake of simplicity, we will often call G a group, without referring to the operation
on it.
Example 3.3. (Z, +) is a group.
Example 3.4. (R, +) is a group. Also: (R \ {0}, ·) is a group.
Example 3.5. (N, +) is not a group!
Remark 3.6. The identity element of a group is often denoted as 1 if the operation
is denoted by the symbol ·, whereas it is denoted as 0 if the operation is denoted by
the symbol +. In a similar way, the inverse element is often denoted as g −1 if the
operation is denoted by the symbol ·, whereas it is denoted as −g if the operation
is denoted by the symbol +.
Definition 3.7 (Abelian group). A group (G, ·) is called abelian if its elements commute
according to the operation ·, i.e. a · b = b · a, ∀ a, b ∈ G.
Often we can consider sets with two operations, like (R, +, ·). If they satisfy some
properties, they are called rings. If they satisfy even more properties, they are called
fields. In particular:
, 4. QUICK REVIEW: MORPHISMS 3
Definition 3.8 (Ring). Given a set R and two operations: + (usually called "additive
operation") and · (called "multiplicative operation") on it, we will call the set with the
two operations, i.e. (R, +, ·), a ring if the following properties are satisfied:
• (R, +) is an abelian group,
• · is associative, i.e. a · (b · c) = (a · b) · c, ∀ a, b, c, ∈ R,
• · is distributive with respect to +, i.e. a·(b+c) = a·b+a·c and (b+c)·a = b·a+c·a.
Definition 3.9 (Commutative ring). (R, +, ·) is a commutative ring if the multiplication
operation · is commutative.
Definition 3.10 (Unitary ring). (R, +, ·) is a unitary ring if it contains the multiplicative
identity, i.e. ∃ 1 ∈ R such that 1 · r = r · 1 = r, ∀ r ∈ R .
Definition 3.11 (Invertible element). An element a in a ring (R, +, ·) is called invertible
if there exists its inverse according to the multiplicative operation, i.e. ∃ b ∈ R such that
a · b = b · a = 1. We will also write a−1 ≡ b.
Definition 3.12 (Field). A unitary, commutative ring such that every non-zero element
is invertible is called a field.
Remark 3.13. When we talk about the zero element in a field or in a ring we always
refer to the identity element according to the additive operation. See also remark
3.6.
Example 3.14. The set of 2 × 2 matrices with real coefficients is a ring with the oper-
ations of sum between matrices and with the matrix multiplication. The ring is not
commutative because the matrix multiplication is not a commutative operation. It
is unitary, since the identity matrix is the multiplicative identity. But it’s not a field
because not every matrix is invertible (just take a matrix with zero determinant).
Example 3.15. (Z, +, ·) is a unitary, commutative ring. It is not a field since the
inverse element for · is often in the rational numbers, i.e. 3−1 is the multiplicative
inverse of 3.
Example 3.16. (R, +, ·) is a field.
4 Quick review: Morphisms
Definition 4.1 (Homomorphism). A homomorphism h between two sets endowed with
structures (think about two groups, for instance) is a map which preserves the structure,
Differential Geometry
October 15, 2022
Contents
Contents I
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Notations and Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
3 Quick review: Basic Algebraic Structures . . . . . . . . . . . . . . . . . . . 2
4 Quick review: Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
5 Quick review: Equivalence Classes, Quotient Spaces . . . . . . . . . . . . . 4
6 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
7 Multilinear Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
8 Exterior Product and Generalisation . . . . . . . . . . . . . . . . . . . . . . 19
9 Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
10 Integration of Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . 32
11 Tangent Bundles, Flows of Vector Fields . . . . . . . . . . . . . . . . . . . . 33
12 Lie Derivative of a Vector Field . . . . . . . . . . . . . . . . . . . . . . . . . 37
13 Stokes’ Theorem on Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
14 Closed and Exact k-forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
15 De Rham Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
16 Submanifolds of Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
17 Integration on Compact Manifolds . . . . . . . . . . . . . . . . . . . . . . . 65
18 Abstract Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
19 Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
20 Compactly Supported Cohomology . . . . . . . . . . . . . . . . . . . . . . . 81
21 Poincaré Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
22 Riemannian Metric and the Hodge-? Operator . . . . . . . . . . . . . . . . . 94
Bibliography 102
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
I
,1. INTRODUCTION 1
1 Introduction
From the Lee book [Lee]: "The central idea of calculus is linear approximation". A function
of one variable can be approximated by its tangent line, a curve by a tangent vector (i.e.
velocity vector), a surface in R3 can be approximated by its tangent plane, and a map from
Rn to Rm by its total derivative. Here it comes the importance of tangent spaces.
Main idea: in order to study tangent vectors, we identify them with "directional deriva-
tives". In particular, there is a natural one-to-one correspondence between geometric tan-
gent vectors and linear maps from C ∞ (Rn ) to R satisfying the product rule. Such maps
are called derivations.
Remark 1.1. Points or vectors? We can think of elements of Rn either as points
or vectors. As points, their only property is their location, given by the coordinates
(x1 , . . . , xn ) on a chosen basis. As vectors, they are characterized by a direction
and a magnitude, but their location is irrelevant (translational invariance). So given
v ∈ Rn , v = i v i ei = v i ei , it can be seen as an arrow with its initial point anywhere
P
in Rn . So, if we think about a vector tangent to the border of the sphere at a point
a, we imagine the vector as living in a copy of Rn with its origin translated to a.
2 Notations and Conventions
In these notes the Einstein summation convention will be used. It means that the sum
symbol will be omitted when it is clear with respect to which index we are summing. For
instance we will write
v i ei
instead of
X
v i ei .
i
Given a map f from a set X to a set Y , we will denote it by
f: X →Y .
We will use the arrow "7→" to denote how each element of X is mapped into Y through f .
For instance, given x1 , x2 ∈ X, y ∈ Y , we will either use the notation:
x1 7→ f (x1 )
or
x2 7→ y .
We will use the symbol ≡ in order to define mathematical objects. We will use the latin
abbreviation "i.e." (id est) to explain concepts in a more detailed way, and the abbreviation
"e.g." (exempli gratia) to give examples.
,3. QUICK REVIEW: BASIC ALGEBRAIC STRUCTURES 2
3 Quick review: Basic Algebraic Structures
Definition 3.1 (Operation). Let G be a set. The map:
· : G × G −→ G
(a, b) 7−→ a · b
is called a (binary) operation on G. Sometimes such a map is also denoted by the symbol
+.
Definition 3.2 (Group). Given a set G and an operation · on such set, we will call such set
with the operation (i.e. the couple (G, ·)) a group if the following properties are satisfied
for all a, b, c ∈ G:
• a · b ∈ G, (closure property)
• a · (b · c) = (a · b) · c, (associativity)
• ∃ 1 ∈ G such that 1 · g = g, ∀ g ∈ G, (existence of the identity element)
• ∀ g ∈ G, ∃ g −1 ∈ G so that g · g −1 = g −1 · g = 1. (existence of the inverse element)
For the sake of simplicity, we will often call G a group, without referring to the operation
on it.
Example 3.3. (Z, +) is a group.
Example 3.4. (R, +) is a group. Also: (R \ {0}, ·) is a group.
Example 3.5. (N, +) is not a group!
Remark 3.6. The identity element of a group is often denoted as 1 if the operation
is denoted by the symbol ·, whereas it is denoted as 0 if the operation is denoted by
the symbol +. In a similar way, the inverse element is often denoted as g −1 if the
operation is denoted by the symbol ·, whereas it is denoted as −g if the operation
is denoted by the symbol +.
Definition 3.7 (Abelian group). A group (G, ·) is called abelian if its elements commute
according to the operation ·, i.e. a · b = b · a, ∀ a, b ∈ G.
Often we can consider sets with two operations, like (R, +, ·). If they satisfy some
properties, they are called rings. If they satisfy even more properties, they are called
fields. In particular:
, 4. QUICK REVIEW: MORPHISMS 3
Definition 3.8 (Ring). Given a set R and two operations: + (usually called "additive
operation") and · (called "multiplicative operation") on it, we will call the set with the
two operations, i.e. (R, +, ·), a ring if the following properties are satisfied:
• (R, +) is an abelian group,
• · is associative, i.e. a · (b · c) = (a · b) · c, ∀ a, b, c, ∈ R,
• · is distributive with respect to +, i.e. a·(b+c) = a·b+a·c and (b+c)·a = b·a+c·a.
Definition 3.9 (Commutative ring). (R, +, ·) is a commutative ring if the multiplication
operation · is commutative.
Definition 3.10 (Unitary ring). (R, +, ·) is a unitary ring if it contains the multiplicative
identity, i.e. ∃ 1 ∈ R such that 1 · r = r · 1 = r, ∀ r ∈ R .
Definition 3.11 (Invertible element). An element a in a ring (R, +, ·) is called invertible
if there exists its inverse according to the multiplicative operation, i.e. ∃ b ∈ R such that
a · b = b · a = 1. We will also write a−1 ≡ b.
Definition 3.12 (Field). A unitary, commutative ring such that every non-zero element
is invertible is called a field.
Remark 3.13. When we talk about the zero element in a field or in a ring we always
refer to the identity element according to the additive operation. See also remark
3.6.
Example 3.14. The set of 2 × 2 matrices with real coefficients is a ring with the oper-
ations of sum between matrices and with the matrix multiplication. The ring is not
commutative because the matrix multiplication is not a commutative operation. It
is unitary, since the identity matrix is the multiplicative identity. But it’s not a field
because not every matrix is invertible (just take a matrix with zero determinant).
Example 3.15. (Z, +, ·) is a unitary, commutative ring. It is not a field since the
inverse element for · is often in the rational numbers, i.e. 3−1 is the multiplicative
inverse of 3.
Example 3.16. (R, +, ·) is a field.
4 Quick review: Morphisms
Definition 4.1 (Homomorphism). A homomorphism h between two sets endowed with
structures (think about two groups, for instance) is a map which preserves the structure,
