Introduction to Differential Geometry
Juan M. Bello-Rivas
bello-rivas@princeton.edu
June 7, 2017
Outline
1 Basic concepts and tools
2 Connections, covariant differentiation, and parallel translation
3 Curvature
Outline
1 Basic concepts and tools
2 Connections, covariant differentiation, and parallel translation
3 Curvature
Smooth manifolds
@
@
@
@R
-
ϕ ψ
ψ ϕ
1
R
m
R
m
Smooth m-dimensional manifold with two overlapping charts.
Smooth maps
Definition
Let M, N be smooth manifolds and let {(U, ϕ)}, {(V, ψ)} be
atlases determining (respectively) the smooth structure of M and
N. A continuous mapping f : M N is said to be a smooth
map, if the composition ψ f ϕ
1
is of class C
.
Smooth maps
A function f : M R is said to be smooth (or of class C
) if it
has “sufficiently” many derivatives (and these are continuous).
The set of all smooth functions from M to N is denoted by
C
(M, N).
The set of all real-valued smooth functions on M is denoted by
C
(M).
Smooth maps
p
f(p)
?
ϕ
6
ψ
1
f
-
-
F
Tangent vectors and tangent spaces
p
T
p
M
M
Tangent vectors and tangent spaces
Definition (Tangent vector)
Let M be a smooth m-dimensional manifold and let p M. A
tangent vector v at p is a map v : C
(M) R such that
1 v(f + g) = v(f) + v(g),
2 v(λf) = λv(f),
3 v(fg) = v(f) g(p) + f (p) v(g),
For any f, g C
(M) and λ R.
Tangent vectors and tangent spaces
The set of tangent vectors at p M is a vector space called the
tangent space, denoted by T
p
M. Indeed, if v, w are tangent
vectors at p and λ R, then
1 (v + w)(f) = v(f) + w(f ),
2 (λ v)(f ) = λ v(f).
In a local chart, the vectors
x
1
p
, . . . ,
x
m
p
constitute a basis of T
p
M.
Tangent vectors and tangent spaces
Example
Let M = R
m
. In this case, the directional derivative of
f C
(M) at p M in the direction v = (v
1
, . . . , v
m
) R
m
is
f(p) · v =
X
i
v
i
f
x
i
(p).
In our current notation, we would instead say that
v =
X
i
v
i
x
i
p
is a tangent vector of M at p.
Tangent vectors and tangent spaces
We can also view each tangent vector at p as the velocity vector of
a curve γ : [0, T ] R M such that γ(0) = p. In that case,
v(f) =
d
dt
f(γ(t)) =
X
i
˙γ
i
(t)
f
x
i
(γ(t))
=
X
i
˙γ
i
(t)
|{z}
=v
i
x
i
γ(t)
(f).
Tangent vectors and tangent spaces
p
T
p
M
M
Vector fields
Definition (Vector field)
A vector field is a correspondence of a tangent vector v
p
to each
point p M. The set of all vector fields over a manifold M is
denoted by X(M ).
In local coordinates,
v
p
=
X
i
v
i
(p)
x
i
p
From now on, we adopt the convention
i
=
x
i
.
Vector fields
To any smooth vector field v X(M) and any t R, we can
associate the flow map, Φ
t
: M M, that satisfies
d
dt
Φ
t
(p) = v
Φ
t
(p)
.
In other words, t 7→ Φ
t
(p) is an integral curve of v.
In local coordinates, if v =
P
i
v
i
i
, then
t 7→ Φ
t
(p) = (γ
1
(t), . . . , γ
m
(t)) solves the initial value problem
d
dt
γ
i
(t) = v
i
(γ(t)),
for each i = 1, . . . , m.
with initial condition Φ
0
(p) = γ(0) = p.
Lie bracket
Definition (Lie bracket)
The Lie bracket of two vector fields v, w X(M ) is the map
[·, ·] : X(M ) × X(M ) X(M)
defined by
(v, w) 7→ [v, w] = vw wv
Proposition
The Lie bracket of two vector fields is a linear differential operator
(the second order derivatives vanish).
Lie bracket
In local coordinates,
[
i
,
j
] =
i
j
j
i
= 0.
The Lie bracket leads us to the generalization of the Theorem of
existence and uniqueness of solutions of ODEs to multiple vector
fields. This general result is known as the Frobenius Theorem.