Differential-Geometry on Chen Kai Blog

Differential Geometry (12): Fiber Bundles, Characteristic Classes, and Physics

Tue, 23 Nov 2021 09:00:00 +0000

Throughout this series, we have built differential geometry from the ground up: manifolds, tangent spaces, differential forms, integration, Riemannian metrics, connections, and curvature. A recurring theme has been the tangent bundle $$TM$$ — the collection of all tangent spaces glued together into a single geometric object. The Levi-Civita connection is a rule for differentiating sections of $$TM$$ (i.e., vector fields), and the Riemann curvature tensor measures the non-commutativity of this differentiation.

But the tangent bundle is just one example of a much more general construction: a fiber bundle. And the Levi-Civita connection is just one example of a connection on a vector bundle. This generalization is not merely aesthetic — it is the mathematical language of gauge theory, the framework underlying all of modern particle physics. Electromagnetism, the weak force, the strong force, and even gravity can all be described as connections on appropriate bundles, with their dynamics governed by curvature.

Differential Geometry (11): Curvature in Riemannian Geometry — Riemann, Ricci, and Scalar

Sun, 21 Nov 2021 09:00:00 +0000

Curvature is the central concept of Riemannian geometry. Intuitively, it measures how much a space deviates from being flat — how parallel lines converge or diverge, how triangles have angle excess or deficit, how volumes grow differently from Euclidean expectations. In the previous article we saw that the path-dependence of parallel transport signals the presence of curvature: a vector carried around a closed loop on $$S^2$$ returns rotated, while on $\mathbb{R}^n$ it does not. The next step is to make this precise, to extract a tensor that exactly captures this rotation, and to understand its various contractions.

Differential Geometry (10): Riemannian Geometry — Metrics, Connections, and Parallel Transport

Fri, 19 Nov 2021 09:00:00 +0000

Up to this point in the series we have studied smooth manifolds with their differentiable structure: charts, tangent vectors, differential forms, exterior calculus, integration. None of this required a notion of distance. We could differentiate functions, integrate top-degree forms, decide whether a distribution is integrable — all without ever measuring how long a curve is or what angle two tangent vectors make. The smooth structure is purely topological-with-derivatives. To do geometry in the classical sense — to measure, to compare, to speak of curvature, to recover the everyday meanings of “length” and “angle” — we need additional structure.

Differential Geometry (9): Integration on Manifolds and Stokes' Theorem

Wed, 17 Nov 2021 09:00:00 +0000

In single-variable calculus, the fundamental theorem says that integrating a derivative over an interval equals the boundary difference: $\int_a^b f'(x)\,dx = f(b) - f(a)$ . The “boundary” of $$[a, b]$$ is the two-point set $\{a, b\}$ , with $$b$$ counted positively and $$a$$ negatively. The right-hand side is the integral of $$f$$ over this signed boundary. The left-hand side is the integral of the derivative over the interval. This is, in essence, every “fundamental theorem” you have ever met — Green’s theorem in the plane, the divergence theorem in three dimensions, the classical Stokes’ theorem on surfaces. They are all instances of one statement on manifolds: the integral of $d\omega$ over $$M$$ equals the integral of $\omega$ over $\partial M$ .

Differential Geometry (8): Differential Forms — The Natural Language of Integration on Manifolds

Mon, 15 Nov 2021 09:00:00 +0000

In vector calculus on $\mathbb{R}^3$ , we have three derivative operations: gradient ( $\nabla f$ ), curl ( $\nabla \times F$ ), and divergence ( $\nabla \cdot F$ ). Each operates on a different type of object (scalar fields, vector fields). Two identities — $\nabla \times (\nabla f) = 0$ and $\nabla \cdot (\nabla \times F) = 0$ — sit there looking like happy coincidences. The three integral theorems (the fundamental theorem for line integrals, the classical Stokes’ theorem, the divergence theorem) appear unrelated and unmotivated except by their statements.

Differential Geometry (7): Vector Fields, Flows, and the Lie Bracket

Sat, 13 Nov 2021 09:00:00 +0000

A tangent vector lives at one point. It tells you “this direction, this speed, right here, right now.” It is fundamentally local — pluck it off the manifold and it remembers nothing about its neighbors. A vector field, by contrast, is what you get when you let one tangent vector at every point conspire smoothly. It is a velocity prescription on the entire manifold: stand anywhere, and the field tells you where to go. Follow the prescription, and you trace out an integral curve. Follow it from every starting point at once, and you get a flow — a one-parameter family of diffeomorphisms that drags the whole manifold along itself like a slow river.

Differential Geometry (6): Smooth Manifolds — Geometry Beyond Embedded Surfaces

Thu, 11 Nov 2021 09:00:00 +0000

The first five chapters of this series lived inside $\mathbb{R}^3$ . We had curves and surfaces, parametrized explicitly, with all the geometric data — first and second fundamental forms, principal curvatures, Christoffel symbols, the Theorema Egregium, Gauss-Bonnet — built up from coordinates we could write down. The Theorema Egregium revealed that the intrinsic story can be told without reference to the embedding. But “without reference to the embedding” still meant “the embedding exists; we just choose not to use it.”

Differential Geometry (5): The Gauss-Bonnet Theorem — Where Geometry Meets Topology

Tue, 09 Nov 2021 09:00:00 +0000

The Theorema Egregium of the previous chapter showed that Gaussian curvature is intrinsic — bend a surface without stretching and $$K$$ does not change. The Gauss-Bonnet theorem, which we develop here, says something equally remarkable in a different direction: integrate $$K$$ over a closed surface and you get a topological invariant. The total curvature of any sphere is $4\pi$ . The total curvature of any torus is $$0$$ . The total curvature of a double torus is $-4\pi$ . These facts are blind to the specific geometry — bend, twist, or smoosh the surface as you please, the total curvature does not change.

Differential Geometry (4): Intrinsic Geometry — Theorema Egregium and Geodesics

Sun, 07 Nov 2021 09:00:00 +0000

The previous two chapters set up a clear dichotomy. Chapter 2 introduced the first fundamental form $\mathrm{I}$ — the intrinsic metric, what an ant on the surface can measure. Chapter 3 introduced the second fundamental form $\mathrm{II}$ and the shape operator — the extrinsic data, how the surface bends in $\mathbb{R}^3$ . From $\mathrm{II}$ we computed Gaussian curvature $K = \det S$ and mean curvature $H = \mathrm{tr}\,S/2$ . By all appearances, both $$K$$ and $$H$$ should depend on the embedding. Bend the surface (without stretching) and you would expect both to change.

Differential Geometry (3): The Shape Operator — Curvature of Surfaces

Fri, 05 Nov 2021 09:00:00 +0000

The previous article gave us the intrinsic apparatus: the first fundamental form $\mathrm{I}$ , encoded as the symmetric matrix $\begin{pmatrix}E & F \\ F & G\end{pmatrix}$ . With it, an ant on the surface can measure lengths, angles, and areas without ever leaving. What an ant on a cylinder cannot do, equipped only with $\mathrm{I}$ , is detect that the cylinder is bent. The cylinder has the same first fundamental form as the plane, yet sits very differently in $\mathbb{R}^3$ .

Differential Geometry (2): Surfaces and the First Fundamental Form: Intrinsic Measurements

Wed, 03 Nov 2021 09:00:00 +0000

Curves were a one-dimensional warm-up. The geometry was governed by ODEs, and a single moving frame caught everything interesting. From now on we go up a dimension and the difficulty rises in three directions at once. Tangent vectors get replaced by tangent planes. The single arc-length parameter splits into two coordinates $$(u, v)$$ , and reparametrization becomes a $2\times 2$ Jacobian matrix instead of a scalar. And — the real change — we acquire two distinct kinds of geometry: intrinsic (what an ant living on the surface can measure) and extrinsic (how the surface bends in the surrounding $\mathbb{R}^3$ ). This article is the intrinsic story. We build the first fundamental form, the $2\times 2$ matrix-valued function that lets the ant measure lengths, angles, and areas without ever leaving the surface.

Differential Geometry (1): Curves in Space — Curvature, Torsion, and the Frenet Frame

Mon, 01 Nov 2021 09:00:00 +0000

I am going to start this series the way every honest course on differential geometry starts: with a single moving particle in $\mathbb{R}^3$ . Not a manifold, not a fiber bundle, not even a surface — just a dot tracing a path. Everything later — Gauss maps, second fundamental forms, connections, Riemannian curvature tensors — is in some sense an effort to do for higher-dimensional objects what we are about to do, very thoroughly, for a one-dimensional one. So bear with the warm-up. The pay-off is that by the end of this article we will own a complete local theory: two scalar functions ( $\kappa$ , $\tau$ ) and one orthonormal frame ( $\mathbf{T}, \mathbf{N}, \mathbf{B}$ ) that together pin down a curve up to rigid motion. The slogan is “curve = two numbers per point”, and that slogan deserves a proof.