MTH 868 — Lecture 07
Date: 2026-01-28
Topic: Regular points, critical points, submanifolds, and regular level sets
1. Setup and Recall
Let
- ( F : M \to N ) be a smooth map between smooth manifolds.
Critical vs Regular Points and Values
-
A critical point ( p \in M ) is one where the differential [ dF_p : T_p M \to T_{F(p)} N ] is not surjective.
-
A regular point is a point where ( dF_p ) is surjective.
- A critical value is a point in ( N ) that is the image of some critical point.
- A regular value is a point in ( N ) that is not a critical value.
Stats intuition:
This is rank deficiency. A critical point is where your Jacobian drops rank.
Regular values are outputs where the Jacobian behaves nicely everywhere in the preimage.
2. Characterizing Critical Points for ( F : M \to \mathbb{R} )
Proposition
Let ( F : M \to \mathbb{R} ) be smooth.
Then ( p \in M ) is a critical point iff:
Equivalent conditions
-
In any chart ( (U, \varphi = (x^1, \dots, x^n)) ) around ( p ), [ \frac{\partial F}{\partial x^i}(p) = 0 \quad \forall i ]
-
There exists a chart where all partial derivatives vanish.
Why this works (lecture explanation)
-
The differential ( dF_p ) is a linear map [ dF_p : T_p M \to T_{F(p)} \mathbb{R} ]
- Since ( \mathbb{R} ) is 1-dimensional, surjectivity means:
- the linear map is non-zero
-
In coordinates, using the basis [ \left{ \frac{\partial}{\partial x^i} \bigg|_p \right} ] the differential is represented by the row vector [ [ \partial_1 F(p) \;\; \cdots \;\; \partial_n F(p) ] ]
- This has rank 0 iff all entries are zero
Key invariant:
“All partial derivatives vanish” is a coordinate-independent statement, even though it looks coordinate-dependent.
3. Why Partial Derivatives Vanishing Is Chart-Independent
Suppose ( (U, x^i) ) and ( (V, y^j) ) are two charts near ( p ).
By the chain rule: [ \frac{\partial F}{\partial y^j} = \sum_i \frac{\partial x^i}{\partial y^j} \frac{\partial F}{\partial x^i} ]
In matrix form: [ [\partial_{y} F] = [\partial_x F] \cdot J(\varphi \circ \psi^{-1}) ]
- The Jacobian of the transition map is invertible
- Therefore: [ [\partial_x F] = 0 \iff [\partial_y F] = 0 ]
Stats analogy:
Gradient = 0 is invariant under reparameterization
Just like score functions transform covariantly.
4. Regular Submanifolds (Definition)
Let ( S \subset N ).
Definition
( S ) is a regular ( k )-dimensional submanifold of ( N ) if:
For every ( p \in S ), there exists a chart
( (U, \varphi = (x^1, \dots, x^n)) ) of ( N ) about ( p ) such that:
[
S \cap U
=
{ q \in U \mid x^{k+1}(q) = \cdots = x^n(q) = 0 }
]
These charts are called adapted charts
Geometric meaning
- Locally, ( S ) looks like ( \mathbb{R}^k )
- The remaining ( n-k ) coordinates are “normal directions”
Important:
Not every chart does this.
The definition is about existence, not universality.
Examples
- Linear subspaces
- ( xy )-plane in ( \mathbb{R}^3 )
- Graphs of smooth functions
Let ( f \in C^\infty(\mathbb{R}) ).
Define:
[
S = { (x, f(x)) } \subset \mathbb{R}^2
]
Define the chart: [ \varphi(x,y) = (x, y - f(x)) ]
Then:
- ( S ) corresponds to ( { y = 0 } )
- Inverse chart: [ \varphi^{-1}(u,v) = (u, v + f(u)) ]
Thus graphs are smooth 1-manifolds.
5. Codimension
Definition
[ \operatorname{codim}(S) = \dim(N) - \dim(S) ]
- Often easier to reason about constraints than dimensions
- Each independent equation increases codimension by 1
6. Level Sets
Definition
Let ( F : N \to M ) be smooth, ( c \in M ).
The ( c )-level set is: [ F^{-1}(c) = { x \in N \mid F(x) = c } ]
Example: Sphere
Let: [ F(x,y,z) = x^2 + y^2 + z^2 ]
Then: [ F^{-1}(1) = S^2 \subset \mathbb{R}^3 ]
Compute differential: [ dF_{(x,y,z)} = [2x \;\; 2y \;\; 2z] ]
- Surjective everywhere except at ( (0,0,0) )
- Since ( (0,0,0) \notin F^{-1}(1) ), 1 is a regular value
7. Implicit Function Theorem View
Let: [ G = F - 1 \quad \Rightarrow \quad F^{-1}(1) = G^{-1}(0) ]
At the north pole ( (0,0,1) ): [ \frac{\partial G}{\partial z} = 2 \neq 0 ]
By the Implicit Function Theorem:
- There exists a neighborhood where [ z = h(x,y) ]
- The level set is locally the graph of a smooth function
Conclusion:
Regular level sets are smooth submanifolds.
8. Regular Level Set Theorem (Main Result)
Theorem
Let ( F : N \to M ) be smooth.
If ( c \in M ) is a regular value, then:
- ( F^{-1}(c) ) is a smooth submanifold
- Dimension: [ \dim F^{-1}(c) = \dim N - \dim M ]
- Codimension: [ \operatorname{codim}(F^{-1}(c)) = \dim M ]
Proof sketch (what actually happened in lecture)
- Reduce to ( c = 0 ) via ( G = F - c )
- Use surjectivity of ( dF_p ) to find a nonzero partial derivative
- Reorder coordinates if needed
- Apply inverse / implicit function theorem
- Show local model is coordinate hyperplane
Stats translation:
Regular value = full-rank Jacobian everywhere on the constraint set
⇒ constraint surface is smooth, dimension reduced by number of constraints
9. Why This Matters (Big Picture)
- This theorem underlies:
- constraint optimization
- Lagrange multipliers
- manifolds defined by equations
- likelihood surfaces under constraints
- parameter identifiability
If you remember one thing:
Regular = full rank Jacobian ⇒ smooth geometry
Everything else is machinery to make that precise.
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