Lecture 6
MTH 868 — Topology / Smooth Manifolds
Date: Jan 26, 2026
Topic: Tangent Vectors, Pushforwards, Curves, Immersions, Submersions, Critical Points
1. Review from Last Time (Page 1)
Smooth Manifolds
We are working with a smooth manifold ( M ).
- Intuition: Locally, ( M ) looks like ( \mathbb{R}^n ).
- Charts allow us to do calculus even though globally the space may be curved or weird.
2. Tangent Space via Derivations (Key Idea)
Definition (Very Important)
The tangent space at a point ( p \in M ), denoted ( T_p M ), is defined as:
The set of derivations at ( p ), i.e., linear maps
[ X : C^\infty(M) \to \mathbb{R} ] satisfying the Leibniz rule: [ X(fg) = f(p)X(g) + g(p)X(f) ]
Stats intuition:
This is an abstract way of defining a directional derivative without coordinates.
Think of it as “how functions change at point ( p ) along some direction”.
Coordinate Description (What Makes This Concrete)
If
[
(U, \phi = (x^1, \dots, x^n))
]
is a chart around ( p ), then the vectors
[
\left{ \frac{\partial}{\partial x^i}\bigg|p \right}{i=1}^n
]
form a basis of ( T_p M ).
Professor emphasis:
“This is why tangent spaces are ( n )-dimensional, not infinite-dimensional.”
Stats parallel:
This is exactly the same reason gradients live in ( \mathbb{R}^n ), even though functions live in infinite-dimensional spaces.
3. Pushforward (Differential of a Map)
Let ( F : M \to N ) be smooth.
Definition
The pushforward (also called the differential) [ F_|_p : T_p M \to T_{F(p)} N ] is defined by [ (F_ X)(f) = X(f \circ F) ]
Interpretation:
- Vectors act on functions
- Pushforward tells you how a vector on ( M ) acts on functions on ( N )
Coordinate Formula (Jacobians Appear!)
If ( F : \mathbb{R}^n \to \mathbb{R}^m ), then [ F_* \left( \frac{\partial}{\partial x^i} \right) = \sum_{j=1}^m \frac{\partial F^j}{\partial x^i} \frac{\partial}{\partial y^j} ]
Professor explicitly said:
“This is the Jacobian. Nothing mysterious is happening.”
Stats connection:
This is exactly the linearization you use in:
- Delta method
- Multivariate Taylor expansions
- Sensitivity analysis
4. Chain Rule for Pushforwards (Proposition)
If [ M \xrightarrow{F} N \xrightarrow{G} P ] are smooth, then [ (G \circ F)*|_p = G|_{F(p)} \circ F_|_p ]
Proof Sketch (from lecture)
For ( X \in T_p M ): [ ((G \circ F)* X)(f) = X(f \circ G \circ F) ] and [ (G* (F_* X))(f) = (F_* X)(f \circ G) = X(f \circ G \circ F) ]
Thus they agree.
Stats analogy:
Linear approximations compose exactly the way you expect.
5. Curves and Velocity Vectors (Page 2)
Definition
A curve in ( M ) is a smooth map [ c : (a,b) \to M ]
Velocity Vector
The velocity at ( t_0 ) is [ c’(t_0) = c_* \left( \frac{d}{dt}\bigg|{t_0} \right) \in T{c(t_0)} M ]
Key intuition:
Tangent vectors are velocities of curves.
Fundamental Proposition
Given any ( p \in M ) and any ( X \in T_p M ),
there exists a curve ( c ) such that
[
c(0) = p, \quad c’(0) = X
]
Professor emphasis:
“This is why curves are not just examples, they characterize tangent vectors.”
Proof Idea (Very Important Construction)
- Choose coordinates ( \phi(p) = 0 )
- Write [ X = \sum a^i \frac{\partial}{\partial x^i} ]
- Define [ c(t) = \phi^{-1}(a^1 t, \dots, a^n t) ]
This is the manifold version of a straight line.
6. Immersions and Submersions (Page 2–3)
Let ( F : M \to N ).
Immersion
( F ) is an immersion if [ F_*|_p \text{ is injective for all } p ]
Meaning:
Locally, ( M ) does not collapse dimensions.
Submersion
( F ) is a submersion if [ F_*|_p \text{ is surjective for all } p ]
Meaning:
Locally, ( F ) hits all directions in ( N ).
Examples
- ( F(t) = (\cos t, \sin t) ) is an immersion
- ( F(t) = (t^2, t^3) ) is not an immersion at ( t=0 )
Why:
Derivative vanishes, rank drops.
7. Rank, Critical Points, Regular Values (Page 3)
Rank
[ \text{rank } F(p) = \dim(\text{image of } F_*|_p) ]
Critical Point
( p \in M ) is a critical point if [ F_*|_p \text{ is not surjective} ]
Regular Point
Otherwise, ( p ) is a regular point.
Regular Value
( y \in N ) is a regular value if [ \forall p \in F^{-1}(y), \; p \text{ is a regular point} ]
Example
[ F(x,y) = x^2 - y^2 ]
- Gradient: [ \nabla F = (2x, -2y) ]
- Only critical point is ( (0,0) )
- Hence:
- Only critical value is ( 0 )
- All other values are regular
Stats analogy:
Critical points are where the Jacobian loses rank.
Regular values are “well-behaved” outputs.
8. Big Picture (Professor’s Framing)
“Everything we do from here on depends on understanding when maps behave like linear maps locally.”
- Immersions → embedded submanifolds
- Submersions → level sets are manifolds
- Regular values → preimages are smooth manifolds
This is the foundation for:
- Implicit Function Theorem
- Sard’s Theorem
- Transversality
- Differential topology
9. Notation Appendix (For Sanity)
| Symbol | Meaning |
|---|---|
| ( T_p M ) | Tangent space at point ( p ) |
| ( F_* ) | Pushforward / differential |
| ( \partial / \partial x^i ) | Coordinate basis vector |
| Immersion | Injective differential |
| Submersion | Surjective differential |
| Regular value | All preimages are regular points |
Comments