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MTH 868 – Lecture 03 (Jan 16, 2026)

Smooth Maps, Diffeomorphisms, and Lie Groups

1. Chain Rule in Several Variables

Let

  • $ f : U \subset \mathbb{R}^m \to \mathbb{R}^n $
  • $ g : V \subset \mathbb{R}^n \to \mathbb{R}^\ell $

Assume:

  • $ f(U) \subset V $
  • $ f $ and $ g $ are smooth ($ C^\infty $)

Then the Jacobian of the composition satisfies:

\[J(g \circ f)(x) = Jg(f(x)) \cdot Jf(x)\]

More explicitly, the $ (i,j) $-entry is: \(\frac{\partial (g \circ f)^i}{\partial x^j} = \sum_{k=1}^n \frac{\partial g^i}{\partial y^k}(f(x)) \frac{\partial f^k}{\partial x^j}(x)\)

Key idea:
The multivariable chain rule is just matrix multiplication of Jacobians.

2. Smooth Manifolds (Reminder)

“Let $ M $ be a manifold”

This means:

  • $ M $ is a topological manifold
  • Equipped with a smooth structure

Formally: A smooth $ n $-manifold $ M $ is a topological space with a maximal smooth atlas \(\{(U_\alpha, \varphi_\alpha)\}_{\alpha \in A}\) where:

  • $ U_\alpha \subset M $ are open
  • $ \varphi_\alpha : U_\alpha \to \mathbb{R}^n $ are homeomorphisms
  • All transition maps
    \(\varphi_\beta \circ \varphi_\alpha^{-1}\) are smooth where defined

3. Smooth Maps Between Manifolds

Definition

Let:

  • $ M $ be an $ m $-manifold with charts $ (U_\alpha, \varphi_\alpha) $
  • $ N $ be an $ n $-manifold with charts $ (V_\beta, \psi_\beta) $

A map $ F : M \to N $ is smooth ($ C^\infty $) if:

For every $ p \in M $,
there exist charts $ U_\alpha \ni p $, $ V_\beta \ni F(p) $ such that:

$$
\psi_\beta \circ F \circ \varphi_\alpha^{-1}
\mathbb{R}^m \to \mathbb{R}^n $$ is smooth as a map of Euclidean spaces.

Important Remark

This definition is independent of the choice of charts.

Why? Because changing charts composes with smooth transition maps.

4. Diffeomorphisms

Definition

A smooth map $ F : M \to N $ is a diffeomorphism if:

  • $ F $ is bijective
  • $ F $ is smooth
  • $ F^{-1} $ is smooth

5. Charts are Local Diffeomorphisms

Let $ (U, \varphi) $ be a chart on a manifold $ M $.

Then: \(\varphi : U \to \varphi(U) \subset \mathbb{R}^n\) is a diffeomorphism onto its image.

Why?

  • $ \varphi $ is a homeomorphism
  • Transition maps guarantee smoothness
  • Identity maps are smooth

Identity map:
\(\text{id}(x) = x\)

6. Example: Torus Map

Recall: \(T^2 = S^1 \times S^1\)

Define: \(F(e^{i\theta_1}, e^{i\theta_2}) = (e^{i(\theta_1 + \theta_2)}, e^{i\theta_2})\)

Then:

  • $ F $ is smooth
  • Inverse is: \(F^{-1}(e^{i\phi_1}, e^{i\phi_2}) = (e^{i(\phi_1 - \phi_2)}, e^{i\phi_2})\)

Hence: \(F : T^2 \to T^2 \text{ is a diffeomorphism}\)

Geometrically: this is a shear on the torus.

7. Diffeomorphisms of $ \mathbb{R} $

Example: \(f : \mathbb{R} \to \mathbb{R}\)

A bijection with smooth inverse is a diffeomorphism.

Fact: Any strictly monotone smooth function with $ f’(x) \neq 0 $ everywhere is a diffeomorphism onto its image.

Counterexample: \(x \mapsto x^3\) is bijective but inverse is not smooth at 0.

8. The Diffeomorphism Group

Define: \(\mathrm{Diff}(M) = \{ F : M \to M \mid F \text{ is a diffeomorphism} \}\)

Then:

  • Closed under composition
  • Inverses exist
  • Identity exists

So $ \mathrm{Diff}(M) $ is a group.

9. Induced Smooth Structures (Important Subtlety)

Let $ Q \subset \mathbb{R}^2 $ be the unit circle.

We know: \(S^1 \subset \mathbb{R}^2\)

But:

  • $ S^1 $ does not inherit a smooth structure by restricting charts of $ \mathbb{R}^2 $

Instead:

  • We give $ S^1 $ a smooth structure using charts modeled on $ \mathbb{R} $

Key point: There is no diffeomorphism: \(\mathbb{R} \to \mathbb{R}^2\) onto a neighborhood of a corner or boundary.

This is why manifolds have a fixed dimension.

10. Smooth Structures up to Diffeomorphism

Being diffeomorphic is an equivalence relation on smooth manifolds.

A diffeomorphism class = one smooth structure up to smooth change of coordinates.

Facts:

  • $ \mathbb{R}^n $ has a unique smooth structure
  • $ S^1 $ has a unique smooth structure
  • But: for $ S^7 \subset \mathbb{R}^8 $, there are multiple smooth structures
    (Milnor’s exotic spheres)

11. Lie Groups

Definition

A Lie group is:

  • A smooth manifold $ M $
  • Equipped with a group structure
  • Such that:
    • Multiplication $ M \times M \to M $ is smooth
    • Inversion $ M \to M $ is smooth

12. Example: $ \mathrm{GL}(n, \mathbb{R}) $

\[\mathrm{GL}(n, \mathbb{R}) = \{ A \in \mathbb{R}^{n \times n} : \det A \neq 0 \}\]

Facts:

  • Open subset of $ \mathbb{R}^{n^2} $
  • Matrix multiplication is polynomial → smooth
  • Inverse map: \(A^{-1} = \frac{1}{\det A} \cdot \text{adj}(A)\) is smooth where $ \det A \neq 0 $

Conclusion: \(\mathrm{GL}(n, \mathbb{R}) \text{ is a Lie group}\)

Mental Map (Big Picture)

  • Charts let you reduce manifold problems to Euclidean calculus
  • Smoothness is defined via charts
  • Diffeomorphisms are the “isomorphisms” of smooth manifolds
  • Lie groups marry algebra + smooth geometry
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