Lecture 5 — Quotients and Tangent Spaces
Date: Jan 23, 2026
Announcements
- Homework due Friday.
- Today: finish quotient spaces, then begin tangent spaces (§8).
1. Review: Lie Group Actions and Quotients
Setup
Let $ G $ be a Lie group, and let $ M $ be a smooth manifold. Assume $ G $ acts smoothly on $ M $: \(G \times M \to M, \quad (g,x) \mapsto g \cdot x\) where each group element acts by a diffeomorphism of $ M $.
Orbit Space / Quotient
Define an equivalence relation on $ M $: \(x \sim y \quad \Longleftrightarrow \quad \exists g \in G \text{ such that } x = g \cdot y.\) The quotient space is \(M/G := M / \sim,\) the set of orbits of the action.
⚠️ Important warning:
In general, $ M/G $ is NOT a smooth manifold. Often it is not even a manifold at all.
2. Example: Complex Projective Space
Let
- $ G = \mathbb{C}^* $ (nonzero complex numbers, multiplicative group),
- $ M = \mathbb{C}^{n+1} \setminus {0} $,
- action: scalar multiplication \(\lambda \cdot (z_0,\dots,z_n) = (\lambda z_0,\dots,\lambda z_n).\)
Then \(\mathbb{CP}^n = (\mathbb{C}^{n+1} \setminus \{0\}) / \mathbb{C}^\*.\)
This quotient does admit a smooth manifold structure, but this is special and requires work.
3. Example: A Quotient That Is NOT a Manifold
Consider:
- $ S^2 \subset \mathbb{R}^3 \cong \mathbb{C} \times \mathbb{R} $,
- $ G = S^1 \subset \mathbb{C} $,
- action: rotation in the complex coordinate \(e^{i\theta} \cdot (z,t) = (e^{i\theta} z, t).\)
What does the quotient look like?
- Each orbit collapses a horizontal circle to a point.
- Only the $ t $-coordinate remains.
- The quotient is homeomorphic to the interval $[-1,1]$.
Why is this NOT a manifold?
- At the north and south poles, the entire group fixes the point.
- The local structure near those points is singular.
- There is no neighborhood homeomorphic to an open interval.
Key lesson:
Quotients by group actions typically produce spaces with singularities. Manifold structure requires strong extra conditions.
4. Motivation: What Is the Tangent Space?
We now turn to §8: Tangent Spaces.
Goal:
Given a smooth manifold $ M $ and a point $ p \in M $, define the tangent space $ T_pM $ intrinsically, without embedding $ M \subset \mathbb{R}^n $.
In Euclidean space, tangent vectors are arrows. On manifolds, we need a coordinate-free definition.
5. Germs of Smooth Functions
Let $ C^\infty(M) $ be the set of smooth real-valued functions on $ M $.
Define an equivalence relation: \(f \sim g \quad \Longleftrightarrow \quad \exists \text{ open } U \ni p \text{ such that } f|_U = g|_U.\)
Definition (Germ)
The equivalence class of $ f $ at $ p $ is called the germ of $ f $ at $ p $.
Denote the set of all germs by: \(C_p^\infty(M).\)
Intuition:
- A germ remembers only the behavior of a function infinitesimally near $ p $.
- Values far away from $ p $ are irrelevant.
Algebra Structure
$ C_p^\infty(M) $ is an $ \mathbb{R} $-algebra:
- addition,
- multiplication,
- scalar multiplication are inherited from functions.
6. Definition: Derivations
Definition
A derivation at $ p $ is a linear map \(D : C_p^\infty(M) \to \mathbb{R}\) satisfying the Leibniz rule: \(D(fg) = D(f)\,g(p) + f(p)\,D(g).\)
This is an abstract version of differentiation.
Definition (Tangent Space)
\(T_pM := \{ \text{derivations at } p \}.\)
7. Tangent Space Is a Vector Space
If $ D_1, D_2 \in T_pM $ and $ \lambda \in \mathbb{R} $, then:
- $ D_1 + D_2 $ is a derivation,
- $ \lambda D_1 $ is a derivation.
Hence $ T_pM $ is a real vector space.
8. Coordinate Description of Tangent Vectors
Let $ (U,\varphi) $ be a chart about $ p $, with: \(\varphi = (x^1,\dots,x^n), \quad \varphi(p)=0.\)
For any vector $ c = (c^1,\dots,c^n) \in \mathbb{R}^n $, define: \(D_c(f) = \sum_{i=1}^n c^i \frac{\partial (f \circ \varphi^{-1})}{\partial x^i}(0).\)
This is a derivation.
Thus we get a linear map: \(\Phi : \mathbb{R}^n \to T_pM, \quad c \mapsto D_c.\)
9. Proposition: $ \Phi $ Is an Isomorphism
Injectivity
If $ c \neq 0 $, then for some coordinate function $ x^i $, \(D_c(x^i) = c^i \neq 0,\) so $ D_c \neq 0 $.
Surjectivity
Given any derivation $ D $, define: \(c^i := D(x^i).\) Using Taylor’s theorem, one shows: \(D = D_c.\)
Thus: \(T_pM \cong \mathbb{R}^n\) (non-canonically, since it depends on the chart).
10. Basis of the Tangent Space
The derivations: \(\left\{ \frac{\partial}{\partial x^1}\Big|_p, \dots, \frac{\partial}{\partial x^n}\Big|_p \right\}\) form a basis for $ T_pM $.
Shorthand: \(\partial_i := \frac{\partial}{\partial x^i}\Big|_p.\)
11. Pushforward of Tangent Vectors
Let $ F : M \to N $ be smooth and $ X \in T_pM $.
Definition (Pushforward)
The pushforward $ F_*X \in T_{F(p)}N $ is defined by: \((F_*X)(g) := X(g \circ F), \quad g \in C^\infty_{F(p)}(N).\)
This defines a linear map: \(F_* : T_pM \to T_{F(p)}N.\)
This generalizes the Jacobian matrix.
Summary
- Quotients by group actions often fail to be manifolds.
- Tangent vectors are defined abstractly as derivations.
- Coordinates recover the familiar partial derivatives.
- Pushforwards generalize Jacobians intrinsically.
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