MTH 868 — Lecture 04
Date: Wednesday, January 21, 2026
Topic: Partial derivatives on manifolds, Jacobians, inverse function theorem, quotient manifolds, projective space
Administrative
- Office hours: M/W 4:05–5:05 PM, D225
Smooth Maps and Charts (Review)
Let $ M, N $ be smooth manifolds.
- A map $ F : M \to N $ is smooth if for all charts \((U,\phi) \text{ on } M, \quad (V,\psi) \text{ on } N,\) the map \(\psi \circ F \circ \phi^{-1} : \phi(U) \subset \mathbb{R}^n \to \mathbb{R}^m\) is smooth in the usual Euclidean sense.
Remark. If $ U \subset M $ is open and $ \phi : U \to \mathbb{R}^n $ is a diffeomorphism onto its image, then $ (U,\phi) $ is a chart.
A maximal atlas contains at least as many charts as $ \mathrm{Diff}(D^n) $, where $ D^n \subset \mathbb{R}^n $.
Coordinate Functions
Let $ (U,\phi) $ be a chart on an $ n $-manifold $ M $, with \(\phi : U \to \mathbb{R}^n.\)
Write \(\phi(p) = (x^1(p), \dots, x^n(p)),\) where the coordinate functions \(x^i := \pi_i \circ \phi, \quad \pi_i : \mathbb{R}^n \to \mathbb{R}\) are smooth functions on $ U $.
Sometimes we denote a chart by \((U, x^1, \dots, x^n).\)
Partial Derivatives on a Manifold
Let $ f : M \to \mathbb{R} $ be smooth and let $ (U,\phi) $ be a chart about $ p \in M $.
Definition (Coordinate Partial Derivative)
The partial derivative of $ f $ at $ p $ in the $ x^i $-direction is \(\frac{\partial f}{\partial x^i}(p) := \left.\frac{\partial}{\partial r_i}\big(f \circ \phi^{-1}\big)\right|_{\phi(p)}.\)
This is the ordinary Euclidean partial derivative of \(f \circ \phi^{-1} : \mathbb{R}^n \to \mathbb{R}.\)
Notation. You may also see: \(\partial_i f(p), \quad \frac{\partial f}{\partial x^i}(p).\)
Important Remark.
This notion of derivative is not intrinsic.
It depends on the choice of chart.
Example: $ S^1 \subset \mathbb{R}^2 $
Let $ S^1 = {(x,y) : x^2 + y^2 = 1} $ and consider the point $ p = (0,1) $.
Define: \(U = \{ (\cos\theta, \sin\theta) : \theta \in (0,\pi) \}, \quad \phi(\cos\theta, \sin\theta) = \theta.\)
Then \(\phi^{-1}(\theta) = (\cos\theta, \sin\theta).\)
Let $ f : S^1 \to \mathbb{R} $ be \(f(x,y) = x.\)
Compute: \(\frac{df}{d\theta}(p) = \left.\frac{d}{d\theta}(\cos\theta)\right|_{\theta=\pi/2} = -\sin(\pi/2) = -1.\)
Geometric intuition. As $ \theta $ increases, the $ x $-coordinate decreases near the north pole.
Remark. The numerical value depends on the chart. Reparametrizing $ \theta $ changes the derivative.
Derivatives of Coordinate Functions
Proposition
If $ (U, x^1, \dots, x^n) $ is a chart, then \(\frac{\partial x^i}{\partial x^j} = \delta^i_j,\) where $ \delta^i_j $ is the Kronecker delta.
Proof
By definition, \(\frac{\partial x^i}{\partial x^j} = \frac{\partial}{\partial r_j}(\pi_i \circ \phi \circ \phi^{-1}) = \frac{\partial}{\partial r_j}(\pi_i) = \delta^i_j.\)
Jacobian of a Smooth Map
Let $ F : N^n \to M^m $ be smooth.
Let:
- $ (U, x^1, \dots, x^n) $ be a chart around $ p \in N $,
- $ (V, y^1, \dots, y^m) $ be a chart around $ F(p) \in M $.
Definition (Jacobian Matrix)
The Jacobian of $ F $ at $ p $ in these coordinates is the matrix \((JF(p))^i_j := \left.\frac{\partial (y^i \circ F)}{\partial x^j}\right|_p.\)
This is exactly the Jacobian of \(\psi \circ F \circ \phi^{-1} : \mathbb{R}^n \to \mathbb{R}^m.\)
Remark. There is no canonical Jacobian of a map between manifolds. It is always coordinate-dependent.
Inverse Function Theorem (Manifold Version)
Theorem
Let $ F : M^n \to N^n $ be smooth.
Then $ F $ is locally invertible (a local diffeomorphism) near $ p \in M $
if and only if there exist charts around $ p $ and $ F(p) $ such that
\(\det(JF(p)) \neq 0.\)
Equivalently, the Jacobian matrix is invertible.
Proof Sketch
Choose charts and reduce to the Euclidean inverse function theorem applied to \(\psi \circ F \circ \phi^{-1}.\)
Equivalence Relations and Quotient Topology
An equivalence relation $ \sim $ on a set $ X $ satisfies:
- Reflexive: $ x \sim x $
- Symmetric: $ x \sim y \Rightarrow y \sim x $
- Transitive: $ x \sim y, y \sim z \Rightarrow x \sim z $
Define equivalence classes: \([x] := \{y \in X : y \sim x\}.\)
The quotient set is: \(X/\!\sim = \{[x] : x \in X\}.\)
If $ X $ is a topological space, the quotient topology on $ X/!\sim $ is: \(U \subset X/\!\sim \text{ open } \iff \pi^{-1}(U) \text{ open in } X,\) where $ \pi : X \to X/!\sim $ is the projection map.
Real Projective Space $ \mathbb{RP}^n $
Definition
\(\mathbb{RP}^n := (\mathbb{R}^{n+1} \setminus \{0\})/\!\sim, \quad x \sim y \iff y = \lambda x \text{ for some } \lambda \neq 0.\)
Interpretation: points correspond to lines through the origin in $ \mathbb{R}^{n+1} $.
Alternative Model
\(\mathbb{RP}^n \cong S^n / (x \sim -x),\) identifying antipodal points on the sphere.
Smooth Structure on $ \mathbb{RP}^n $
Let \([a^0 : \dots : a^n]\) denote an equivalence class.
For each $ i = 0,\dots,n $, define: \(U_i = \{[a^0:\dots:a^n] : a^i \neq 0\}.\)
Define charts \(\phi_i : U_i \to \mathbb{R}^n\) by \(\phi_i([a^0:\dots:a^n]) = \left( \frac{a^0}{a^i}, \dots, \widehat{\frac{a^i}{a^i}}, \dots, \frac{a^n}{a^i} \right),\) omitting the $ i $-th coordinate.
Properties
- Well-defined (scaling cancels)
- Smooth
- Transition maps are smooth rational functions
Thus $ \mathbb{RP}^n $ is a smooth $ n $-manifold.
Complex Projective Space
Similarly, \(\mathbb{CP}^n := (\mathbb{C}^{n+1} \setminus \{0\})/\!\sim, \quad z \sim \lambda z, \ \lambda \in \mathbb{C}^\times.\)
- Charts defined analogously
- Transition maps are holomorphic
- $ \mathbb{CP}^n $ is a complex manifold of complex dimension $ n $
Quotients by Group Actions
Let a group $ G $ act smoothly on a manifold $ M $.
Define: \(x \sim y \iff \exists g \in G \text{ such that } x = g \cdot y.\)
Then \(M/G := M/\!\sim\) is a quotient space.
Example
Let $ G = \mathbb{R}^\times $ act on $ \mathbb{R}^{n+1} \setminus {0} $ by scalar multiplication. Then \(\mathbb{RP}^n \cong (\mathbb{R}^{n+1} \setminus \{0\}) / \mathbb{R}^\times.\)
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