MTH 868 – Lecture 1 Expanded Notes
(Topology / Differential Geometry for Manifolds)
0. Big Picture (Why This Course Looks Like This)
This course is building the analytic foundations of smooth manifolds.
The first few weeks look like multivariable real analysis because:
- Manifolds are locally Euclidean
- Smooth structure is defined via smooth maps between open sets of ℝⁿ
- Everything reduces locally to statements about Jacobians, derivatives, and rank
If you understand:
- multivariable differentiation,
- inverse / implicit function theorems,
- Jacobians and rank,
then the rest of differential geometry is mostly bookkeeping plus geometry.
1. Smooth Maps Between Euclidean Spaces
Definition: Vector-Valued Functions
A function
\(F : \mathbb{R}^n \to \mathbb{R}^m\)
is written componentwise as
\(F(x) = (F^1(x), F^2(x), \dots, F^m(x))\)
where each $F^i:\mathbb{R}^n\to\mathbb{R}$.
This viewpoint is critical because all derivatives are taken componentwise.
2. Jacobian and Differential
Jacobian Matrix
At a point $p\in\mathbb{R}^n$, the Jacobian of $F$ is the matrix \(JF(p) = \begin{pmatrix} \frac{\partial F^1}{\partial x^1} & \cdots & \frac{\partial F^1}{\partial x^n} \\ \vdots & \ddots & \vdots \\ \frac{\partial F^m}{\partial x^1} & \cdots & \frac{\partial F^m}{\partial x^n} \end{pmatrix}\)
Differential
The differential \(DF_p : \mathbb{R}^n \to \mathbb{R}^m\) is the linear map represented by $JF(p)$.
Interpretation:
$DF_p$ is the best linear approximation to $F$ at $p$.
In probability/statistics terms:
- Think first-order Taylor expansion
- Everything geometric later depends on this linearization
3. Smoothness ($C^n$ and $C^\infty$)
Multi-Index Notation
A multi-index $\alpha=(\alpha_1,\dots,\alpha_n) $ has order \(\|\alpha\| = \alpha_1 + \cdots + \alpha_n\)
Mixed partial derivative: \(\partial^\alpha F^i = \frac{\partial^{\|\alpha\|} F^i} {\partial x_1^{\alpha_1} \cdots \partial x_n^{\alpha_n}}\)
Definition: Smooth ($C^\infty$)
A function $F$ is smooth if all partial derivatives of all orders exist and are continuous.
Smoothness is the default regularity class for manifolds in this course.
4. Local Diffeomorphisms
Definition
Let $U\subset\mathbb{R}^n$ be open.
A smooth map $F:U\to\mathbb{R}^n$ is a local diffeomorphism at $x\in U$ if:
- There exist neighborhoods $V\ni x$, $W\ni F(x)$
- $F\vert_V:V\to W$ is bijective
- The inverse $F^{-1}:W\to V$ is smooth
This is a local notion, not global.
5. Inverse Function Theorem (IFT)
Statement
Let $U\subset\mathbb{R}^n$ be open and
$F:U\to\mathbb{R}^n$ be $C^\infty$.
If \(\det(JF(x)) \neq 0\) then $F$ is a local diffeomorphism at $x$.
Interpretation
- Nonzero determinant = derivative is invertible
- Locally, $F$ behaves like a linear isomorphism
- Geometry near $x$ is preserved (up to smooth distortion)
6. Important Examples (Why Determinant ≠ 0 Matters)
Example 1: $ f(x) = x^3 $
- Smooth and bijective on ℝ
- $f’(0) = 0$
- Not a local diffeomorphism at 0
- Inverse derivative blows up
Lesson: Global bijection does NOT imply local diffeomorphism.
Example 2: $g(z)=z^2$ on $\mathbb{C}\cong\mathbb{R}^2$
- Smooth
- Jacobian determinant $=0$ at $z=0$
- Not injective (many square roots)
Lesson: Rank failure kills invertibility.
7. Implicit Function Theorem (IFT – Geometric Version)
Setup
Let \(F : \mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^m\) and suppose \(F(x_0, y_0) = 0\)
We want to solve: \(F(x, y) = 0\)
Intuition
- Expect an $ n $-dimensional solution set
- Want to write solutions as $y=h(x)$
Statement
If \(\det\left( \frac{\partial F}{\partial y}(x_0, y_0) \right) \neq 0\) then:
- There exist neighborhoods $A\ni x_0$, $B\ni y_0$
- There exists a unique smooth function $h:A\to B$
- Such that \(F(x, h(x)) = 0\)
8. Example: Circle as an Implicit Curve
Let \(F(x, y) = x^2 + y^2 - 1\)
- Zero set = unit circle
- Near points where $y\neq 0$, solve for $y=\pm\sqrt{1-x^2}$
- Near points where $x\neq 0$, solve for $x$ instead
Key Insight:
Manifolds are often not global graphs, but always local graphs.
9. Manifolds (Chapter 2 Start)
Motivation
A manifold is a space that:
- Looks like ℝⁿ locally
- May be curved or topologically nontrivial globally
Definition: Smooth n-Manifold
A topological space $ M $ is a smooth n-dimensional manifold if:
(1) Topological Conditions
- Hausdorff
- Second countable
(Technical conditions ensuring good behavior of limits, partitions of unity, etc.)
(2) Locally Euclidean
For every $p \in M$, there exists:
- An open set $U\subset M$
- A homeomorphism (chart) \(\varphi : U \to \varphi(U) \subset \mathbb{R}^n\)
This means locally: \(M \approx \mathbb{R}^n\)
(3) Smooth Compatibility
If $(U, \varphi)$ and $(V, \psi)$ are charts with overlap:
\[\psi \circ \varphi^{-1} : \varphi(U \cap V) \to \psi(U \cap V)\]must be smooth.
These are called transition maps.
10. Why This Matters for You (Stats Perspective)
- Tangent spaces = derivatives on manifolds
- Parameter spaces in statistics are manifolds
- Constraints define manifolds via implicit function theorem
- Fisher information is a Riemannian metric
- Optimization on manifolds uses these tools directly
11. What You Should Review (Minimal Survival Kit)
If you are missing prerequisites, focus ONLY on:
- Multivariable derivatives and Jacobians
- Inverse Function Theorem
- Implicit Function Theorem
- Linear algebra: rank, determinant, invertibility
You do not need:
- Abstract algebra
- Algebraic topology (yet)
- Homology theory (yet)
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