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Differential Geometry

The study of curves, surfaces, and higher-dimensional manifolds using the tools of calculus and linear algebra — from classical Frenet-Serret theory to the Gauss-Bonnet theorem and modern manifold theory.

Curves in R³ Surfaces Geodesics Gauss-Bonnet Manifolds Riemannian Metrics
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01 Curves in $\mathbb{R}^3$

Space curves are studied via their curvature and torsion, which together (up to rigid motion) completely determine a curve. The Frenet-Serret formulas give the rate of change of the moving frame.

◆ Curvature and Torsion
Definition — Regular Curve A smooth curve $\alpha: I \to \mathbb{R}^3$ is regular if $\alpha'(t) \ne 0$ for all $t \in I$. The arc-length parameter $s$ satisfies $\|\alpha'(s)\| = 1$.
Definition — Curvature & Torsion For a unit-speed curve $\alpha(s)$:
Curvature: $\kappa(s) = \|\alpha''(s)\| \ge 0$.
Torsion: $\tau(s) = -\langle N'(s), B(s)\rangle$ where $N = \frac{\alpha''}{\kappa}$ (principal normal) and $B = T \times N$ (binormal), with $T = \alpha'$.

For a general parametrisation $\alpha(t)$:

  • $\kappa = \dfrac{\|\alpha' \times \alpha''\|}{\|\alpha'\|^3}$
  • $\tau = \dfrac{(\alpha' \times \alpha'') \cdot \alpha'''}{\|\alpha' \times \alpha''\|^2}$
◆ Frenet-Serret Formulas
Theorem — Frenet-Serret Equations For a unit-speed curve with $\kappa > 0$, the Frenet frame $\{T, N, B\}$ satisfies: $$\begin{pmatrix} T' \\ N' \\ B' \end{pmatrix} = \begin{pmatrix} 0 & \kappa & 0 \\ -\kappa & 0 & \tau \\ 0 & -\tau & 0 \end{pmatrix} \begin{pmatrix} T \\ N \\ B \end{pmatrix}.$$
Theorem — Fundamental Theorem of Space Curves Given smooth functions $\kappa(s) > 0$ and $\tau(s)$, there exists a unit-speed curve in $\mathbb{R}^3$ with curvature $\kappa$ and torsion $\tau$, unique up to rigid motions (rotations and translations).
★ Example
Compute the curvature and torsion of the circular helix $\alpha(t) = (a\cos t,\; a\sin t,\; bt)$ where $a > 0$.
Solution: $\alpha' = (-a\sin t, a\cos t, b)$, $\|\alpha'\| = \sqrt{a^2 + b^2} := c$. $\alpha'' = (-a\cos t, -a\sin t, 0)$, $\alpha''' = (a\sin t, -a\cos t, 0)$. Then $\alpha' \times \alpha'' = (ab\sin t, -ab\cos t, a^2)$ with $\|\alpha' \times \alpha''\| = a\sqrt{a^2+b^2} = ac$. So $\kappa = \frac{ac}{c^3} = \frac{a}{a^2+b^2}$ (constant). Also $(\alpha' \times \alpha'') \cdot \alpha''' = a^2 b$, giving $\tau = \frac{a^2 b}{a^2 c^2} = \frac{b}{a^2+b^2}$ (constant). A helix has constant curvature and constant torsion.
02 Surfaces

Surfaces in $\mathbb{R}^3$ are studied through their first and second fundamental forms, which encode intrinsic and extrinsic geometry respectively.

◆ First & Second Fundamental Forms
Definition — First Fundamental Form For a regular parametrised surface $\mathbf{r}(u,v)$, the first fundamental form is: $$\mathrm{I} = E\,du^2 + 2F\,du\,dv + G\,dv^2$$ where $E = \langle \mathbf{r}_u, \mathbf{r}_u \rangle$, $F = \langle \mathbf{r}_u, \mathbf{r}_v \rangle$, $G = \langle \mathbf{r}_v, \mathbf{r}_v \rangle$. It measures lengths, angles, and areas on the surface.
Definition — Second Fundamental Form With unit normal $\mathbf{n} = \frac{\mathbf{r}_u \times \mathbf{r}_v}{\|\mathbf{r}_u \times \mathbf{r}_v\|}$, the second fundamental form is: $$\mathrm{II} = L\,du^2 + 2M\,du\,dv + N\,dv^2$$ where $L = \langle \mathbf{r}_{uu}, \mathbf{n}\rangle$, $M = \langle \mathbf{r}_{uv}, \mathbf{n}\rangle$, $N = \langle \mathbf{r}_{vv}, \mathbf{n}\rangle$.
◆ Gaussian & Mean Curvature
Definition — Principal, Gaussian & Mean Curvature The principal curvatures $\kappa_1, \kappa_2$ are the eigenvalues of the shape operator (Weingarten map) $S = \mathrm{I}^{-1}\mathrm{II}$. Then:
Gaussian curvature: $K = \kappa_1 \kappa_2 = \dfrac{LN - M^2}{EG - F^2}$.
Mean curvature: $H = \dfrac{\kappa_1 + \kappa_2}{2} = \dfrac{EN - 2FM + GL}{2(EG - F^2)}$.
Theorem — Gauss's Theorema Egregium The Gaussian curvature $K$ depends only on the first fundamental form (i.e., it is an intrinsic invariant). Consequently, $K$ is preserved under isometries of surfaces.
★ Example
Compute the Gaussian curvature of the sphere $\mathbf{r}(\theta, \phi) = (R\sin\theta\cos\phi,\; R\sin\theta\sin\phi,\; R\cos\theta)$.
Solution: $E = R^2$, $F = 0$, $G = R^2\sin^2\theta$. The unit normal is $\mathbf{n} = \frac{1}{R}\mathbf{r}$ (outward). $L = R$, $M = 0$, $N = R\sin^2\theta$. Then $K = \frac{LN - M^2}{EG - F^2} = \frac{R \cdot R\sin^2\theta}{R^2 \cdot R^2\sin^2\theta} = \frac{1}{R^2}$ (constant, as expected for a sphere of radius $R$).
03 Geodesics

Geodesics are the “straightest” curves on a surface — curves whose geodesic curvature vanishes. They generalise straight lines to curved spaces.

◆ Geodesic Equations
Definition — Geodesic A curve $\gamma(t)$ on a surface is a geodesic if its acceleration vector is always normal to the surface, i.e., the covariant derivative $\frac{D\gamma'}{dt} = 0$. In local coordinates $(u(t), v(t))$, this gives the system: $$u'' + \Gamma_{11}^1 u'^2 + 2\Gamma_{12}^1 u'v' + \Gamma_{22}^1 v'^2 = 0$$ $$v'' + \Gamma_{11}^2 u'^2 + 2\Gamma_{12}^2 u'v' + \Gamma_{22}^2 v'^2 = 0$$ where $\Gamma_{jk}^i$ are the Christoffel symbols of the first fundamental form.
Definition — Geodesic Curvature The geodesic curvature of a curve $\gamma$ on a surface is $\kappa_g = \kappa \cos\theta$ where $\theta$ is the angle between the principal normal of $\gamma$ and the tangent plane. A geodesic satisfies $\kappa_g = 0$ everywhere.
★ Example
Find the geodesics on a cylinder $x^2 + y^2 = R^2$.
Solution: Parametrise the cylinder as $\mathbf{r}(u,v) = (R\cos u, R\sin u, v)$. Then $E = R^2$, $F = 0$, $G = 1$, and all Christoffel symbols vanish. The geodesic equations become $u'' = 0$ and $v'' = 0$, giving $u = at + b$, $v = ct + d$. These trace helices (including lines parallel to the axis when $a = 0$ and circles when $c = 0$). Since the cylinder is isometric to the plane, geodesics correspond to straight lines on the unrolled plane.
04 Gauss-Bonnet Theorem

The Gauss-Bonnet theorem is one of the deepest results in differential geometry, connecting local curvature to global topology.

◆ Local & Global Gauss-Bonnet
Theorem — Local Gauss-Bonnet Let $R$ be a simply connected region on a surface bounded by a piecewise smooth curve $\partial R$ with exterior angles $\theta_1, \ldots, \theta_k$ at vertices. Then: $$\int_{\partial R} \kappa_g\,ds + \iint_R K\,dA + \sum_{i=1}^{k} \theta_i = 2\pi.$$
Theorem — Global Gauss-Bonnet For a compact orientable surface $S$ without boundary: $$\iint_S K\,dA = 2\pi\,\chi(S)$$ where $\chi(S)$ is the Euler characteristic. For a surface of genus $g$, $\chi = 2 - 2g$.
★ Example
Verify the Gauss-Bonnet theorem for the sphere $S^2$ of radius $R$.
Solution: $K = \frac{1}{R^2}$ (constant), $\text{Area}(S^2) = 4\pi R^2$, and $\chi(S^2) = 2$ (genus $0$). Then $\iint K\,dA = \frac{1}{R^2} \cdot 4\pi R^2 = 4\pi = 2\pi \cdot 2 = 2\pi\chi(S^2)$. Verified.
05 Introduction to Manifolds

Manifolds generalise curves and surfaces to arbitrary dimensions, providing the geometric framework for modern physics and advanced geometry.

◆ Smooth Manifolds & Tangent Spaces
Definition — Smooth Manifold An $n$-dimensional smooth manifold is a Hausdorff, second-countable topological space $M$ equipped with an atlas $\{(U_\alpha, \phi_\alpha)\}$ of charts $\phi_\alpha: U_\alpha \to \mathbb{R}^n$ such that all transition maps $\phi_\beta \circ \phi_\alpha^{-1}$ are smooth ($C^\infty$).
Definition — Tangent Space The tangent space $T_pM$ at $p \in M$ is the vector space of derivations on germs of smooth functions at $p$. In local coordinates $(x^1, \ldots, x^n)$, $T_pM$ has basis $\left\{\frac{\partial}{\partial x^1}\Big|_p, \ldots, \frac{\partial}{\partial x^n}\Big|_p\right\}$.
◆ Differential Forms
Definition — Differential $k$-Form A differential $k$-form on $M$ is a smooth section of $\Lambda^k(T^*M)$. In local coordinates, a $k$-form is $\omega = \sum_{i_1 < \cdots < i_k} f_{i_1\ldots i_k}\,dx^{i_1} \wedge \cdots \wedge dx^{i_k}$. The exterior derivative $d: \Omega^k(M) \to \Omega^{k+1}(M)$ satisfies $d^2 = 0$.
Theorem — Stokes' Theorem (on Manifolds) For an oriented smooth $n$-manifold $M$ with boundary and a compactly supported $(n-1)$-form $\omega$: $$\int_M d\omega = \int_{\partial M} \omega.$$
◆ Riemannian Metrics
Definition — Riemannian Metric A Riemannian metric on a smooth manifold $M$ is a smooth assignment of an inner product $g_p: T_pM \times T_pM \to \mathbb{R}$ to each point $p \in M$. In local coordinates: $g = \sum_{i,j} g_{ij}\,dx^i \otimes dx^j$ where $g_{ij} = g\!\left(\frac{\partial}{\partial x^i}, \frac{\partial}{\partial x^j}\right)$.

A Riemannian metric defines lengths of curves ($L(\gamma) = \int \sqrt{g(\gamma', \gamma')}\,dt$), angles between tangent vectors, volumes ($dV = \sqrt{\det(g_{ij})}\,dx^1 \wedge \cdots \wedge dx^n$), and a distance function making $(M, d)$ a metric space.

★ Example
Write the Riemannian metric for the hyperbolic plane (upper half-plane model).
Solution: The upper half-plane $\mathbb{H}^2 = \{(x,y) \in \mathbb{R}^2 : y > 0\}$ with the metric $g = \frac{dx^2 + dy^2}{y^2}$ has $g_{11} = g_{22} = \frac{1}{y^2}$, $g_{12} = 0$. The Gaussian curvature is $K = -1$ (constant negative curvature). Geodesics are vertical lines and semicircles with centres on the $x$-axis.
★ Key Takeaways
✎ Practice Problems
Problem 1
Show that a curve $\alpha(s)$ in $\mathbb{R}^3$ (parametrised by arc length) is a plane curve if and only if $\tau(s) = 0$ for all $s$.
Show Solution ▼
If $\tau = 0$, then $B' = -\tau N = 0$, so $B$ is constant. Thus $\langle \alpha - \alpha(0), B \rangle' = \langle T, B \rangle = 0$, so $\alpha$ lies in the plane $\langle x - \alpha(0), B \rangle = 0$. Conversely, if $\alpha$ lies in a plane with normal $\mathbf{n}$, then $\langle T, \mathbf{n} \rangle = 0$ and differentiating gives $\kappa\langle N, \mathbf{n}\rangle = 0$. If $\kappa > 0$, then $N \perp \mathbf{n}$, so $B = T \times N = \pm\mathbf{n}$ is constant, hence $\tau = 0$.
Problem 2
Compute the Gaussian curvature of the surface of revolution $\mathbf{r}(u,v) = (f(u)\cos v, f(u)\sin v, g(u))$ where $f > 0$ and $(f')^2 + (g')^2 = 1$.
Show Solution ▼
With the unit-speed condition $(f')^2 + (g')^2 = 1$, we get $E = 1$, $F = 0$, $G = f^2$. Computing the second fundamental form: $L = f'g'' - f''g'$, $M = 0$, $N = fg'$. Since $(f')^2 + (g')^2 = 1$, differentiation gives $f'f'' + g'g'' = 0$, so $L = f'g'' - f''g' = -(f''g'^2 + f''f'^2)/(g') \cdot g'/1$. More directly, $K = \frac{LN - M^2}{EG - F^2} = \frac{(f'g'' - f''g') \cdot fg'}{f^2}$. Using $f'f'' + g'g'' = 0$ and simplifying: $K = -\frac{f''}{f}$.
Problem 3
Show that great circles are geodesics on the sphere $S^2$.
Show Solution ▼
A great circle is the intersection of $S^2$ with a plane through the origin. Parametrise as $\gamma(t) = (\cos t)\,\mathbf{p} + (\sin t)\,\mathbf{q}$ where $\mathbf{p}, \mathbf{q}$ are orthonormal vectors. Then $\gamma''(t) = -\gamma(t)$, which is along the normal to the sphere (since the outward normal at $\gamma$ is $\gamma$ itself for the unit sphere). Thus the tangential component of $\gamma''$ vanishes, i.e., $\kappa_g = 0$, so $\gamma$ is a geodesic.
Problem 4
Using the Gauss-Bonnet theorem, compute $\iint_T K\,dA$ for the torus $T$ (genus $1$).
Show Solution ▼
The torus has genus $g = 1$, so $\chi(T) = 2 - 2(1) = 0$. By the global Gauss-Bonnet theorem, $\iint_T K\,dA = 2\pi\chi(T) = 0$. This makes geometric sense: the outer part of the torus has $K > 0$ and the inner part has $K < 0$, and they cancel exactly.
Problem 5
Let $\omega = x\,dy \wedge dz + y\,dz \wedge dx + z\,dx \wedge dy$ be a 2-form on $\mathbb{R}^3$. Compute $d\omega$ and verify Stokes' theorem for the unit ball.
Show Solution ▼
$d\omega = dx \wedge dy \wedge dz + dy \wedge dz \wedge dx + dz \wedge dx \wedge dy = 3\,dx \wedge dy \wedge dz$. By Stokes: $\int_B d\omega = 3\,\text{Vol}(B) = 3 \cdot \frac{4\pi}{3} = 4\pi$. On the boundary $S^2$, $\omega$ restricted to $S^2$ with outward normal $\mathbf{n} = (x,y,z)$ gives $\int_{S^2} \omega = \int_{S^2} (x^2 + y^2 + z^2)\,dA = \int_{S^2} 1\,dA = 4\pi$. Both sides agree.
🎯 Interactive Quiz
1. A circular helix has constant curvature $\kappa$ and constant torsion $\tau$. What is the curvature of a circle (planar curve)?
A $\kappa = 0$
B $\kappa = 1/R$, $\tau = 0$
C $\kappa = R$, $\tau = 0$
D $\kappa = 1/R$, $\tau = 1/R$
2. Gauss's Theorema Egregium states that:
A Mean curvature is an intrinsic invariant
B Gaussian curvature depends only on the first fundamental form
C Principal curvatures are intrinsic
D Total curvature equals $2\pi\chi$
3. The Gaussian curvature of a flat torus (obtained by identifying opposite sides of a rectangle) is:
A $K = 0$ everywhere
B $K > 0$ on the outside, $K < 0$ on the inside
C $K = 1$ everywhere
D $K = -1$ everywhere
4. By the Gauss-Bonnet theorem, the total Gaussian curvature $\iint K\,dA$ for a compact orientable surface of genus $2$ is:
A $4\pi$
B $0$
C $-4\pi$
D $-2\pi$
5. The geodesics on a cylinder $x^2 + y^2 = R^2$ are:
A Only circles
B Only straight lines parallel to the axis
C Helices (including lines and circles as special cases)
D Only non-closing spirals