Differential Geometry

Formulas

Let us take a thread bent in space. How can its shape be conveyed mathematically? The usual parameterization $r(t) = (x(t), y(t), z(t))$ depends on an arbitrary choice of parameter $t$—which is inconvenient. We need invariants: numbers describing the shape which do not depend on the choice of coo...

For a curve, there are two such invariants: curvature (how quickly the tangent turns) and torsion (how the tangent plane rotates around the curve). Together they fully determine the shape of the curve—and it is precisely owing to them that the Frenet frame is constructed: a moving orthonormal bas...

Let us switch to natural parameterization—by arc length $s$. Definition: $s = \int_0^t |r'(\tau)| d\tau$, whence $ds/dt = |r'(t)|$. In natural parameterization, the speed is constant: $|r'(s)| = 1$. This is “unit-length speed”—motion along the curve at a constant speed of $1$ (meter per meter).

Why is natural parameterization important? Because derivatives with respect to $s$ have geometric meaning: $r'(s)$ is the unit tangent, $r''(s)$ shows how quickly the curve “turns”.

Definitions

Evolute

— the locus of centers of curvature of the curve r(s): E(s) = r(s) + (1/κ(s)) ν(s).

Involute

— the reverse concept: a curve for which the given one is the evolute. Geometrically: if you unwind a thread wound around the evolute, the end of the thread describes the involute.

Formulas

The length of a curve is the first and most fundamental metric concept. For a curve r(t) = (x(t), y(t), z(t)) on [a, b]:

Invariance: Length does not depend on parameterization. If we substitute t = φ(u) (φ' > 0): L is preserved—that is the geometric meaning of length.

Variational principle: Among all curves in space between two points, the shortest is the straight segment. This is a minimization problem for the functional L[r] = ∫|r'| dt. On a sphere, the shortest is an arc of a great circle. On a surface—the geodesic.

Numerical example: Spiral r(t) = (cos t, sin t, t) over t ∈ [0, 2π]: r' = (−sin t, cos t, 1). |r'| = √(sin²t + cos²t + 1) = √2. Length: L = 2π√2 ≈ 8.886. For comparison: circle r = 1 gives L = 2π ≈ 6.28. The spiral is longer although it projects onto the same circle.

Formulas

The shape of a plane curve (without regard to its position in space) is completely determined by the dependence of curvature κ on arc length s: natural equation κ = κ(s). If this equation is set, the shape of the curve is determined (up to translations and reflections).

This is a powerful idea: instead of coordinates (x(s), y(s)), a compact functional condition κ(s). Restoring the curve: tangent angle θ(s) = ∫₀ˢ κ(t) dt, then x(s) = ∫cos θ ds, y(s) = ∫sin θ ds.

Coordinates via Fresnel integrals: x(s) = ∫₀ˢ cos(t²/2) dt, y(s) = ∫₀ˢ sin(t²/2) dt.

Geometry: As s → 0: curve → straight line (κ = 0). As s → ∞: the curve spirals toward a finite point — the “Cornu focus”.

Definitions

The most important example: the Mercator projection

— a conformal mapping of the sphere onto the plane (1569). Angles are preserved $\to$ navigational courses are straight lines. But areas are distorted: Greenland appears as large as Africa, though in reality it is 14 times smaller.

Formulas

Imagine a flat sheet of paper. It can be rolled into a cylinder or a cone. For a creature living on the sheet (an ant), the flat sheet and the cylinder are indistinguishable — distances, angles, and areas remain unchanged. This is intrinsic geometry — the geometry “seen” by a surface dweller.

The first fundamental form encodes precisely the intrinsic geometry: how to measure distances and areas while living on the surface, without reference to the external 3D space.

A surface is given by a smooth mapping $r(u, v) = (x(u,v), y(u,v), z(u,v))$ from the parameter domain $(u, v)$ into $\mathbb{R}^3$.

Tangent vectors: $r_u = \partial r/\partial u = (\partial x/\partial u, \partial y/\partial u, \partial z/\partial u),\ r_v = \partial r/\partial v$. At each point they generate the tangent plane $T_pM$.

Definitions

Principal directions

— directions in which normal curvature attains extreme values. These are the eigenvectors of the shape matrix (the matrix II relative to I).

Formulas

The first fundamental form describes the "internal life" of a surface — distances for a resident living on it. The second fundamental form describes how the surface bends in the surrounding three-dimensional space — what an external observer sees.

Intuition: a cylinder and a plane have the same "internal" geometry (they can be mapped isometrically onto each other by "unfolding" the cylinder). But they look different from outside: the cylinder is curved in ℝ³, the plane is not. The second form captures this "external" bending.

Here r_{uu} = ∂²r/∂u², n is the unit normal to the surface. The meaning of L: "how quickly the normal deviates when moving along the u-direction".

Meusnier’s Theorem: the normal curvature depends only on the direction (u', v'), but not on the particular curve in that direction.

In 1827, Gauss published a result that he himself called "Theorema Egregium" — "Remarkable Theorem". It is stated as follows: Gaussian curvature K is an intrinsic property of the surface.

What does this mean? K is computed solely through the coefficients of the first fundamental form E, F, G and their derivatives — without reference to the ambient space. Although the formula $K = (LN - M^2)/(EG - F^2)$ uses the second fundamental form (which is "extrinsic"), Gauss showed that by s...

Consequence: if two surfaces are isometric (can be identified preserving distances), they have the same K in corresponding points. Plane: K = 0 ↔ cylinder: K = 0 — isometric ✓. Plane: K = 0 ↔ sphere: K = $1/R^2$ — not isometric → no map without distortion.

Gauss's formula (Brioschi–Bourquet–Gauss): K can be expressed via Christoffel symbols, which themselves are expressed in terms of E, F, G.

A surface in $\mathbb{R}^3$ is an intuitive object: we “see” it from the outside. But many natural geometric objects are not embedded in familiar space. The phase space of a mechanical system with $n$ degrees of freedom is a $2n$-dimensional manifold. The space of quantum states is an infinite-di...

A smooth manifold is a way to work with such spaces without referencing an external embedding.

Definition: A Hausdorff topological space $M$ in which every point has a neighborhood homeomorphic (topologically equivalent) to an open ball in $\mathbb{R}^n$ is called an $n$-dimensional topological manifold.

Simply put: a manifold is “locally flat”—near every point it looks like a piece of $\mathbb{R}^n$. A surface is “flat” like $\mathbb{R}^2$, space is “flat” like $\mathbb{R}^3$, although globally it may be more complicated.

Formulas

On a surface in ℝ³, a tangent vector is literally “an arrow lying on the surface.” But how do you define a tangent vector to an abstract manifold M not embedded in ℝⁿ?

An elegant solution: a tangent vector at point p is an equivalence class of smooth curves γ: (−ε, ε) → M with γ(0) = p (with the same speed in local coordinates). Or, equivalently, a differential operator v(f) = (f ∘ γ)'(0) — acting on functions (derivative along γ).

The second definition always works and reveals: “vector” = “a way to differentiate functions at a point.”

TₚM is the vector space of all tangent vectors to M at point p. The dimension dim(TₚM) = n.

Definitions

First fundamental form of a surface

— an example of Riemannian metric in two dimensions.

Geodesics

— “straight lines” on a Riemannian manifold: $\nabla_{\gamma'} \gamma' = 0$. In coordinates: $\gamma''^i + \Gamma^i_{jk} \gamma'^j \gamma'^k = 0$ — an ODE system for $\gamma(t)$.

Formulas

A smooth manifold $M$ by itself does not have notions of length or angle—it is “amorphous” topology. The Riemannian metric $g$ is an additional structure that defines an inner product in each tangent space.

Formally: $g$ is a symmetric, positive-definite $(0,2)$-tensor on $M$. At each point $p$: $g_p: T_pM \times T_pM \to \mathbb{R}$ — an inner product. In coordinates: $g = g_{ij} dx^i \otimes dx^j$, symmetry $g_{ij} = g_{ji}$, $g > 0$.

Length of a curve $\gamma$: $L(\gamma) = \int \sqrt{g(\gamma', \gamma')} dt = \int \sqrt{g_{ij} \gamma'^i \gamma'^j} dt$.

Distance: $d(p, q) = \inf_{\gamma: p \to q} L(\gamma)$ — the shortest distance along curves.

The standard integral ∫∫_D f dA “lives” in the plane. How do we integrate on a curved surface? We need an object that transforms correctly under a change of coordinates — a differential form.

An n-form ω on an n-dimensional manifold M is a “density” for integration: in coordinates, ω = f(x) dx¹ ∧ ... ∧ dxⁿ. When changing coordinates, the Jacobian of the transformation is “built in” automatically — the integral is invariant.

A manifold M is orientable if there exists an atlas with positive Jacobians of the transition functions: det(∂x/∂y) > 0. Orientation is the choice of a “consistent” direction in all charts.

On an orientable M, a “right-handed” basis is chosen at each point. This allows us to define a “positive” volume element and the integral.

Formulas

From $d^2 = 0$ it follows: exact form ($\omega = d\eta$) $\rightarrow$ closed ($d\omega = 0$). But the converse does not always hold: closed $\nrightarrow$ exact. This "gap" is connected with the topology of the manifold—the presence of "holes."

Intuition: on the punctured plane $\mathbb{R}^2 \setminus \{0\}$ the form $\omega = \frac{x\,dy - y\,dx}{x^2 + y^2}$ is closed ($d\omega = 0$), but $\oint_{|r| = 1} \omega = 2\pi \ne 0$. If $\omega = df$, then $\oint \omega = 0$. Therefore, $\omega$ is not exact—the "hole" at the origin obstructs...

This is a deep idea: topological properties of space (holes, "handles") are detected by analytic means (integration of forms).

Elements of $H^k$ are equivalence classes of forms up to addition of an exact form. If $\omega_1$ and $\omega_2$ differ by an exact form, they represent the same class.

Definitions

$SU(2) \cong S^3$

— the 3-sphere! Element: $U = aI + i(b\sigma_1 + c\sigma_2 + d\sigma_3)$, $|a|^2 + |b|^2 + |c|^2 + |d|^2 = 1$. $SU(2) \to SO(3)$—a double covering (twofold covering: $\pm U \to$ one rotation).

The symmetries of physical systems—rotations, translations, gauge transformations—form not just groups, but continuous families of transformations. These objects are called Lie groups—smooth manifolds equipped with a group structure.

The key idea of Lie (1870s): continuous symmetries are described by “infinitesimal” transformations—elements of the Lie algebra. Instead of studying the entire group (a complex nonlinear object), we study its algebra (a linear space!) and “reconstruct” the group via the exponential.

A Lie group is a smooth manifold $G$ with operations: multiplication $(a, b) \mapsto ab$ and inversion $a \mapsto a^{-1}$, both smooth.

$GL(n, \mathbb{R})$—invertible $n\times n$ matrices, an open subset of $M(n, \mathbb{R}) \cong \mathbb{R}^{n^2}$, $\dim = n^2$.

Formulas

Analysis on $\mathbb{R}$ or $\mathbb{R}^n$ uses a metric (distance) to define continuity, open sets, convergence. But many important spaces do not have a "reasonable" metric: the space of all continuous functions on $[0,1]$ with pointwise topology, the space of measures, spaces in functional anal...

Topology gives the minimal structure for defining continuity: it suffices to know which sets are "open" — without the concept of distance.

Definition: A pair $(X, \tau)$, where $X$ is a set, $\tau \subset 2^X$ is a family of "open" sets satisfying: $\emptyset, X \in \tau$; the union of any family from $\tau$ is in $\tau$; the intersection of any finite family from $\tau$ is in $\tau$.

Metric topology: $U$ is open $\leftrightarrow$ for all $x \in U$ there exists $B(x, \varepsilon) \subset U$. The standard topology of $\mathbb{R}^n$.

Formulas

A connected space cannot be split into two non-empty, disjoint open sets. Formally: X is connected if X = U ∪ V, U ∩ V = ∅, U, V open ⇒ U = ∅ or V = ∅.

Intuition: a connected space is “all in one piece.” A disconnected space is “broken up” into parts.

Examples: ℝ is connected; removing one point makes it disconnected: ℝ {0} = (−∞, 0) ∪ (0, +∞). The union (0,1) ∪ (2,3) is disconnected (an explicit division into two open sets). The irrational numbers ℚ^c are disconnected as a subspace of ℝ. GL(2,ℝ) is disconnected: matrices with det > 0 and det...

Component of connectedness: The maximal connected subset. For disconnected X: X = ⊔_α C_α. The number of components β₀ is a topological invariant.

A metric space $(X, d)$ is a set $X$ with a distance function $d: X \times X \rightarrow \mathbb{R}_{\ge 0}$ satisfying: $d(x,y) = 0 \iff x = y$; $d(x,y) = d(y,x)$; $d(x,z) \le d(x,y) + d(y,z)$ (the triangle inequality).

$p$-adic: $|n|_p = p^{-v_p(n)}$, where $v_p$ is the exponent of $p$ in the factorization of $n$. Used in number theory and cryptography.

$(X, d)$ is complete if every Cauchy sequence ($d(x_m, x_n) \to 0$ as $m,n \to \infty$) converges to an element of $X$.

Examples: $\mathbb{R}$ is complete (completeness axiom). $\mathbb{Q}$ is not complete: the sequence $3, 3.1, 3.14, 3.141, \ldots$ is Cauchy in $\mathbb{Q}$ but converges to $\pi \notin \mathbb{Q}$. $C[a,b]$ with the uniform norm is complete (Banach space).

Formulas

Take a torus (the surface of a bagel). Wind a thread around a "hole" (along one of the two independent cycles). Can this thread be "shrunk" to a point while staying on the torus? No! But on a sphere, any loop can be contracted.

The fundamental group π₁(X, x₀) formalizes this observation: it counts the "number of ways to wind a loop" which cannot be reduced to each other by a continuous deformation.

This is a topological invariant: homeomorphic spaces have isomorphic fundamental groups. Different π₁ → not homeomorphic.

A loop at a point x₀ ∈ X: a continuous map γ: [0,1] → X with γ(0) = γ(1) = x₀.

Take the spiral ℝ and "wrap" it into the circle S¹: p(t) = e^{2πit}. Each point z ∈ S¹ corresponds to infinitely many pre-images in ℝ (sheets): p⁻¹(z) = {n + arg(z)/2π : n ∈ ℤ}. Locally, ℝ looks "like" S¹, but globally ℝ "covers" S¹ with infinitely many sheets.

A continuous mapping p: X̃ → X is a covering if for every point x ∈ X there exists an open neighborhood U ∋ x (elementary neighborhood), such that p⁻¹(U) = ⊔_α Ũ_α (disjoint union), and each Ũ_α is homeomorphic to U via p.

X̃ is the covering space, X is the base space. The fiber p⁻¹(x) is a discrete subset of X̃. The number of sheets |p⁻¹(x)| = deg(p) is the degree of the covering.

S¹ → S¹: pₙ(z) = zⁿ. n-sheeted covering. Fiber: p⁻¹(1) = {e^{2πik/n} : k = 0,...,n−1}.

Formulas

It turns out that compact closed surfaces (2-manifolds without boundary) are subject to a complete classification: each of them is homeomorphic to exactly one surface from a unique "list". This is one of the jewels of algebraic topology—a rare case where a classification problem is solved complet...

Why is this beautiful? In dimensions 3 and higher, classification is fundamentally impossible (algorithmically undecidable). Dimension 2 is exceptional!

Orientable (genus = g handles): $S^2$ ($g=0$, sphere), $T^2$ ($g=1$, torus), $T^2 \# T^2$ ($g=2$, double torus), ..., $\#^g T^2$ ($g$ handles, $g \geq 0$).

Nonorientable (k projective planes): $\mathbb{R}P^2$ ($k=1$), $K$ ($k=2$, Klein bottle), $\mathbb{R}P^2 \# \mathbb{R}P^2 \# \mathbb{R}P^2$ ($k=3$), ..., $\#^k \mathbb{R}P^2$ ($k \geq 1$).