Manifolds
The mathematical language of curved spaces. From Earth's surface to Einstein's spacetime, manifolds are everywhere. Let's build your intuition from scratch.
Your Journey to Understanding Manifolds
Before we dive into definitions, let's realize something surprising: you've been living on a manifold your entire life!
Earth's Surface
Standing on Earth, everything looks flat - you can walk in any direction, measure distances, draw maps. But we know Earth is actually a sphere!
This is the essence of a manifold: locally flat, globally curved.
Why "Manifold"?
The word comes from German "Mannigfaltigkeit" (many-foldedness), coined by Bernhard Riemann in 1854. It captures the idea of many local pieces folded together into a global shape.
The Big Idea
A manifold is a space where you can do calculus - take derivatives, integrate, find shortest paths - even though the space might be curved. The trick? Work locally where things look flat, then piece together the global picture.
Definition (Informal)
An n-dimensional manifold is a space where every point has a neighborhood that looks like ordinary n-dimensional Euclidean space ℝⁿ.
Breaking It Down
- •n-dimensional: You need n numbers to specify a location. A surface is 2D (needs 2 coordinates), a curve is 1D (needs 1 coordinate).
- •Every point: This property holds everywhere on the manifold, not just at special points.
- •Neighborhood: A small region around the point. How small depends on the point - some regions might need to be very small!
- •Looks like ℝⁿ: There's a continuous, invertible mapping between the neighborhood and flat Euclidean space.
Manifolds
- ✓ Sphere (S²)
- ✓ Torus (donut)
- ✓ Möbius strip
- ✓ Klein bottle
- ✓ Flat plane (ℝ²)
- ✓ Any smooth curve
NOT Manifolds
- ✗ Figure-8 (crossing point)
- ✗ Cone tip (singular point)
- ✗ Two planes meeting at a line
- ✗ Graph with branches
- These have "bad points" that don't look like ℝⁿ
Definition (Formal)
topological space that is locally homeomorphic to ℝⁿ.
Hausdorff: Any two distinct points can be separated by neighborhoods.
Second-countable: Has a countable basis (technical condition for "nice" behavior).
Locally homeomorphic to ℝⁿ: Every point has a neighborhood that can be continuously mapped to ℝⁿ with a continuous inverse.
This is the most important idea to internalize: zoom in far enough on any manifold, and it looks flat. Play with the demo below to see this in action!
You Can See the Curvature
From this distance, the surface clearly curves away from you. The grid lines bend, and you can tell you're on a sphere, not a flat plane. This is the global view.
Earth Example
Earth is a 2D manifold (sphere). Standing on it, everything looks flat - that's the local view. But zoom out to space, and you see it's curved. Ancient people thought Earth was flat because they only saw the local patch!
Why This Matters
This property lets us do calculus on manifolds! Since every small patch looks like flat space, we can use all our familiar tools (derivatives, integrals) locally, then piece together the global picture.
Why This Matters
This property is what makes manifolds so useful:
- 1.Calculus works: Since small patches look like ℝⁿ, we can take derivatives and integrals just like in ordinary calculus.
- 2.Local coordinates: We can use familiar (x, y, ...) coordinates in small regions, even on curved spaces.
- 3.Physics applications: Laws of physics can be written locally, then stitched together to describe global behavior on curved spacetime.
Manifolds come in amazing variety. Click on each one to learn about its unique properties!
Circle
1D manifold
S¹
The simplest closed 1D manifold. Locally looks like a line segment, globally forms a loop.
Real World
Clock faces, angles, periodic phenomena (daily cycles)
Euler Characteristic
χ = 0
A topological invariant: χ = V - E + F
Fun Fact
The circle is the only 1D compact, connected manifold without boundary!
Explore different manifolds in 3D! Drag to rotate, or let them auto-spin. Notice how each surface has its own unique geometry.
Drag to rotate manually
2-Sphere (S²)
2D surface in 3D space
The surface of a ball. Every point looks the same locally - like a tiny flat patch. It's compact (finite size) and has no boundary.
Key Property
Positive curvature everywhere
How do we work with curved spaces mathematically? We use charts - local coordinate systems that map pieces of the manifold to flat space. A collection of charts covering the whole manifold is called an atlas.
What is a Chart?
A chart is a mapping from a piece of the manifold to flat Euclidean space (ℝⁿ). It's like a map of a city - you can't flatten the whole Earth onto one map without distortion, but you can make accurate flat maps of small regions.
An atlas is a collection of charts that together cover the entire manifold, just like a world atlas contains many maps that together show all of Earth.
The Map Analogy
Just like you can't make a perfectly accurate flat map of Earth (some distortion is inevitable), you can't cover a curved manifold with a single coordinate system. But you CAN make many overlapping maps that together show everything!
Transition Functions
Where charts overlap, we need rules for converting between coordinate systems. These "transition functions" must be smooth (differentiable) for a smooth manifold - this is what allows us to do calculus!
Formal Definition
Atlas: Collection of charts {(Uₐ, φₐ)} where ∪Uₐ = M
Transition: φᵦ ∘ φₐ⁻¹ : φₐ(Uₐ ∩ Uᵦ) → φᵦ(Uₐ ∩ Uᵦ)
A smooth manifold requires all transition functions to be smooth (C∞).
On a curved surface, we can't just add vectors like in flat space - "forward" at different points means different things! The tangent space at a point is the set of all possible velocities at that point.
The Point p
Any point on the manifold. At each point, we can ask: "What directions can I move while staying on the surface?"
Tangent Space TₚM
The set of ALL possible velocities at point p. It's a flat vector space that "touches" the manifold at exactly one point - like a plane touching a sphere.
Tangent Vectors
Vectors in the tangent space represent velocities or directions. They're the "derivatives" on manifolds - essential for physics and differential geometry!
Why Tangent Spaces Matter
On a curved surface, you can't just add vectors like in flat space - a velocity pointing "forward" at the North Pole means something different than at the Equator! The tangent space gives us a local flat space where we CAN do vector math, then we translate results back to the manifold.
// Tangent space has same dimension as the manifold
Key Concepts
Tangent Vector
A tangent vector at point p represents a direction and speed you could move while staying on the manifold. Mathematically, it's an equivalence class of curves passing through p with the same velocity.
Tangent Bundle
The collection of ALL tangent spaces at ALL points forms the tangent bundle TM - itself a manifold of dimension 2n (for an n-manifold)!
Why Tangent Spaces Matter
- • Physics: Velocity, momentum, and force are tangent vectors
- • Optimization: Gradients live in tangent spaces
- • Machine Learning: Neural network updates follow tangent directions
- • Robotics: Robot configurations form manifolds; velocities are tangent vectors
Curvature measures how a manifold deviates from being flat. There are different types of curvature, but Gaussian curvature is special: it's intrinsic, meaning beings living on the surface can measure it without knowing about the surrounding space!
Positive Curvature (K > 0)
Example: Sphere - like Earth's surface
Triangles: Triangle angles sum to MORE than 180°
Parallel Lines
Parallel lines eventually meet (like longitude lines at poles)
Gaussian Curvature
// Product of principal curvatures
Theorema Egregium (Gauss's "Remarkable Theorem"): Gaussian curvature is intrinsic - it can be measured by beings living on the surface without knowing about the surrounding 3D space! This is why you can't flatten a sphere without distortion, but you CAN unroll a cylinder (K = 0) into a flat sheet.
K > 0 (Positive)
Like a sphere. Parallel lines converge. Triangles have angle sum > 180°. Walking in a "straight line" eventually brings you back!
K = 0 (Zero)
Flat plane or cylinder. Parallel lines stay parallel. Familiar Euclidean geometry. Can be "unrolled" without distortion.
K < 0 (Negative)
Like a saddle. Parallel lines diverge. Triangles have angle sum < 180°. More space than you'd expect (hyperbolic geometry).
Theorema Egregium
Gauss's "Remarkable Theorem": Gaussian curvature is intrinsic! This means:
- • A 2D being could measure K without knowing about 3D space
- • You can't change K by bending without stretching
- • This is why you can't flatten a sphere (K > 0) or make an accurate flat world map
- • But you CAN unroll a cylinder (K = 0) into a flat rectangle!
General Relativity
Einstein's greatest insight: gravity IS the curvature of spacetime!Spacetime is a 4D manifold, and mass/energy curve it. Objects follow geodesics (shortest paths) on this curved manifold.
Robotics
A robot's configuration space is often a manifold. A robot arm with 6 joints lives on a 6D manifold (not ℝ⁶ because of constraints). Planning motion means finding paths on this manifold!
SO(3) - the rotation group - is a 3D manifold crucial for robotics
Machine Learning
The "manifold hypothesis": real-world high-dimensional data often lies on lower-dimensional manifolds. Images of faces form a manifold in pixel space. This insight powers dimensionality reduction and generative models.
t-SNE, UMAP, autoencoders all leverage manifold structure
Computer Graphics
3D models are 2D manifolds! Mesh processing, texture mapping, and shape analysis all use manifold mathematics. Games with "wrapping" worlds (like Pac-Man) are actually tori!
The surface of any solid 3D object is a 2-manifold
Quantum Mechanics
Quantum states live on complex projective spaces (manifolds!). Berry phase, topological insulators, and quantum computing all use manifold geometry. The Bloch sphere for qubits is S².
State space for n qubits: ℂP^(2ⁿ-1)
Statistics
Probability distributions on a sample space form a manifold (statistical manifold). Information geometry studies this using Riemannian metrics. The Fisher information matrix defines curvature!
Leads to natural gradient descent in optimization
Smooth Manifold (Formal)
a maximal smooth atlas A = {(Uₐ, φₐ)} where all transition
functions φᵦ ∘ φₐ⁻¹ are smooth (C∞) diffeomorphisms.
Tangent Space (Formal)
where γ₁ ~ γ₂ iff (φ ∘ γ₁)'(0) = (φ ∘ γ₂)'(0) for some chart φ
Tangent vectors are equivalence classes of curves with the same velocity at p.
Riemannian Metric
A smoothly varying inner product on each tangent space,
allowing us to measure lengths, angles, and volumes.
A manifold with a Riemannian metric is called a Riemannian manifold. This is the setting for most of differential geometry and general relativity.
Gaussian Curvature
where κ₁, κ₂ are principal curvatures (max and min
normal curvatures at a point)
For surfaces: K > 0 (sphere-like), K = 0 (flat/cylinder), K < 0 (saddle-like).
Euler Characteristic
Sphere: χ = 2
Torus: χ = 0
Klein bottle: χ = 0
ℝP²: χ = 1
A topological invariant! No matter how you deform the surface (without tearing), χ stays the same. Connected to curvature via Gauss-Bonnet: ∫∫ K dA = 2πχ
Manifolds Quiz
Question 1 of 10What is the defining property of a manifold?
You Now Understand Manifolds!
From locally flat spaces to curved spacetime, you've journeyed through the mathematics that describes the shape of our universe. Here's what you've learned:
Manifolds are locally flat but globally curved
We map curved spaces to flat coordinates
Velocities and bending on curved surfaces