Differential Geometry – Complete Reference Guide

Curves • Surfaces • Curvature • Geodesics • Fundamental Forms • Gauss-Bonnet Theorem

Parametric Curves

Definition

A curve $\alpha: I \to \mathbb{R}^3$ is a smooth map from an interval $I$ to space.

$\alpha(t) = (x(t), y(t), z(t))$

Regular Curve

$\alpha'(t) \neq 0$ for all $t$ (non-vanishing velocity)

Allows for well-defined tangent vector

Arc Length

$s(t) = \int_{t_0}^{t} \|\alpha'(u)\| \, du$

Arc length parametrization: $\|\alpha'(s)\| = 1$

Denoted by parameter $s$

Key: Arc length is geometric (independent of parametrization).

Frenet-Serret Frame

The Moving Frame {T, N, B}

Vector	Definition	Name
$\mathbf{T}$	$\frac{\alpha'(s)}{\\|\alpha'(s)\\|}$	Unit Tangent
$\mathbf{N}$	$\frac{\mathbf{T}'(s)}{\\|\mathbf{T}'(s)\\|}$	Unit Normal
$\mathbf{B}$	$\mathbf{T} \times \mathbf{N}$	Binormal

Frenet-Serret Formulas

$\mathbf{T}' = \kappa \mathbf{N}$

$\mathbf{N}' = -\kappa \mathbf{T} + \tau \mathbf{B}$

$\mathbf{B}' = -\tau \mathbf{N}$

Memory: $\kappa$ controls bending, $\tau$ controls twisting.

Curvature & Torsion

Curvature $\kappa$

$\kappa = \|\mathbf{T}'(s)\| = \frac{\|\alpha' \times \alpha''\|}{\|\alpha'\|^3}$

Measures rate of change of tangent direction

$\kappa \geq 0$ always

Radius of curvature: $R = \frac{1}{\kappa}$

Torsion $\tau$

$\tau = -\mathbf{B}' \cdot \mathbf{N} = \frac{(\alpha' \times \alpha'') \cdot \alpha'''}{\|\alpha' \times \alpha''\|^2}$

Measures twisting out of osculating plane

$\tau = 0$ ⟺ curve is planar

$\tau > 0$: right-handed twist

Special Curves

Line: $\kappa = 0$

Circle: $\kappa = \frac{1}{r}$, $\tau = 0$

Helix: $\kappa, \tau$ both constant

Osculating Objects

Osculating Plane

Plane spanned by $\mathbf{T}$ and $\mathbf{N}$

Best-fitting plane to curve at point

Normal vector: $\mathbf{B}$

Osculating Circle

Circle of curvature with radius $R = 1/\kappa$

Center at $\alpha(s) + R \cdot \mathbf{N}$

Lies in osculating plane

Osculating Sphere

Radius: $\rho = \sqrt{R^2 + (R'/\tau)^2}$

Best-fitting sphere to curve

Parametric Surfaces

Definition

$\mathbf{r}: U \subset \mathbb{R}^2 \to \mathbb{R}^3$

$\mathbf{r}(u,v) = (x(u,v), y(u,v), z(u,v))$

Regular Surface

$\mathbf{r}_u \times \mathbf{r}_v \neq 0$ everywhere

Well-defined tangent plane

Tangent Plane

Spanned by $\mathbf{r}_u$ and $\mathbf{r}_v$

Normal: $\mathbf{n} = \frac{\mathbf{r}_u \times \mathbf{r}_v}{\|\mathbf{r}_u \times \mathbf{r}_v\|}$

Common Parametrizations

Surface	Parametrization
Sphere	$(R\sin\phi\cos\theta, R\sin\phi\sin\theta, R\cos\phi)$
Cylinder	$(R\cos\theta, R\sin\theta, z)$
Torus	$((R+r\cos v)\cos u, ...)$

First Fundamental Form

Definition (Metric Tensor)

$I = E\,du^2 + 2F\,du\,dv + G\,dv^2$

Coefficients

Coeff.	Formula
$E$	$\mathbf{r}_u \cdot \mathbf{r}_u$
$F$	$\mathbf{r}_u \cdot \mathbf{r}_v$
$G$	$\mathbf{r}_v \cdot \mathbf{r}_v$

What It Measures

Arc Length: $ds^2 = E\,du^2 + 2F\,du\,dv + G\,dv^2$

Area: $dA = \sqrt{EG - F^2}\,du\,dv$

Angle: $\cos\theta = \frac{F}{\sqrt{EG}}$

Intrinsic: First fundamental form is intrinsic—measurable by "beings" living on surface.

Second Fundamental Form

Definition

$II = L\,du^2 + 2M\,du\,dv + N\,dv^2$

Coefficients

Coeff.	Formula
$L$	$\mathbf{r}_{uu} \cdot \mathbf{n}$
$M$	$\mathbf{r}_{uv} \cdot \mathbf{n}$
$N$	$\mathbf{r}_{vv} \cdot \mathbf{n}$

What It Measures

How surface bends in ambient space

Rate of change of normal vector

Extrinsic: Second fundamental form depends on embedding—not intrinsic!

Shape Operator (Weingarten)

Definition

$S: T_pM \to T_pM$

$S(\mathbf{v}) = -D_\mathbf{v}\mathbf{n}$

Matrix Representation

$S = \begin{pmatrix} E & F \\ F & G \end{pmatrix}^{-1} \begin{pmatrix} L & M \\ M & N \end{pmatrix}$

Properties

Self-adjoint (symmetric)

Eigenvalues = principal curvatures

Eigenvectors = principal directions

Principal Curvatures

Definition

$\kappa_1, \kappa_2$ = eigenvalues of shape operator $S$

Normal Curvature

$\kappa_n(\theta) = \kappa_1 \cos^2\theta + \kappa_2 \sin^2\theta$

Euler's formula for curvature in direction $\theta$

From Fundamental Forms

$\kappa_n = \frac{II}{I} = \frac{L\,du^2 + 2M\,du\,dv + N\,dv^2}{E\,du^2 + 2F\,du\,dv + G\,dv^2}$

Principal Directions

Eigenvectors of shape operator

Directions of max/min normal curvature

Always orthogonal (if $\kappa_1 \neq \kappa_2$)

Umbilic Point: $\kappa_1 = \kappa_2$ (every direction is principal).

Gaussian & Mean Curvature

Gaussian Curvature $K$

$K = \kappa_1 \kappa_2 = \det(S) = \frac{LN - M^2}{EG - F^2}$

$K > 0$: elliptic point (bowl-like)

$K < 0$: hyperbolic point (saddle-like)

$K = 0$: parabolic point

Mean Curvature $H$

$H = \frac{\kappa_1 + \kappa_2}{2} = \frac{1}{2}\text{tr}(S)$

$H = \frac{EN - 2FM + GL}{2(EG-F^2)}$

$H = 0$: minimal surface

Surface Classification

Type	$K$
Sphere	$K = 1/R^2 > 0$
Cylinder	$K = 0$
Saddle	$K < 0$

Theorema Egregium

Gauss's Remarkable Theorem

Gaussian curvature $K$ is intrinsic!

$K$ depends only on $E, F, G$ and their derivatives

Does NOT depend on how surface sits in space

Consequences

Can't flatten sphere without distortion

Maps preserve $K$ ⟺ isometries

Cylinder ($K=0$) can flatten to plane

Intuition: A bug on a surface can measure $K$ using only intrinsic measurements!

Geodesics

Definition

Curve with geodesic curvature $\kappa_g = 0$

Acceleration always normal to surface

Locally shortest path

Geodesic Equations

$\ddot{u}^k + \Gamma^k_{ij}\dot{u}^i\dot{u}^j = 0$

$\Gamma^k_{ij}$ = Christoffel symbols

Examples

Straight lines on plane

Great circles on sphere

Helices on cylinder

Geodesic Curvature

$\kappa_g = \kappa \sin\psi$ where $\psi$ = angle with normal

Decomposition: $\kappa^2 = \kappa_g^2 + \kappa_n^2$

Christoffel Symbols

Definition (First Kind)

$\Gamma_{ijk} = \frac{1}{2}(g_{ik,j} + g_{jk,i} - g_{ij,k})$

Second Kind

$\Gamma^k_{ij} = g^{kl}\Gamma_{ijl}$

Properties

6 independent symbols for surfaces

Symmetric: $\Gamma^k_{ij} = \Gamma^k_{ji}$

Role

Define covariant derivative

Appear in geodesic equation

Connect metric to curvature

Covariant Derivative

Definition

$\nabla_X Y = (X(Y^k) + \Gamma^k_{ij}X^i Y^j)\partial_k$

Properties

$\nabla_X(Y+Z) = \nabla_X Y + \nabla_X Z$

$\nabla_{fX} Y = f\nabla_X Y$

$\nabla_X(fY) = X(f)Y + f\nabla_X Y$

$\nabla g = 0$ (metric compatible)

Parallel Transport

$\nabla_{\dot{\gamma}} V = 0$ along curve $\gamma$

Preserves length and angle

Path-dependent (related to curvature)

Gauss-Bonnet Theorem

Local Version (Polygon)

$\iint_R K\,dA + \int_{\partial R} \kappa_g\,ds + \sum \epsilon_i = 2\pi$

$\epsilon_i$ = exterior angles at vertices

Global Version (Compact)

$\iint_M K\,dA = 2\pi\chi(M)$

$\chi(M)$ = Euler characteristic

Euler Characteristic

$\chi = V - E + F$

Surface	$\chi$
Sphere	$2$
Torus	$0$
g-holed	$2-2g$

Amazing: Total curvature is a topological invariant!

Minimal Surfaces

Definition

Surface with $H = 0$ (zero mean curvature)

Characterizations

Critical points of area functional

$\kappa_1 = -\kappa_2$ at every point

Local area minimizers

Examples

Plane: Simplest minimal surface

Catenoid: Rotation of catenary

Helicoid: Twisted surface

Enneper: Self-intersecting

Isometries & Mappings

Isometry

Map preserving first fundamental form

Preserves distances and angles

Preserves Gaussian curvature

Conformal Map

Preserves angles but not distances

$E' = \lambda E$, $F' = \lambda F$, $G' = \lambda G$

Developable Surface

$K = 0$ everywhere

Can flatten to plane without stretching

Examples: cones, cylinders

DIFFERENTIAL GEOMETRY Complete Reference