5. Introduction to Differential Geometry

5.1. Differential geometry in \(E^3\)

5.1.1. Curves

Parametric representation of curve in 3-dimensional (Euclidean) space \(E^3\)

\[\vec{r}(q)\]

Differential, \(d \vec{r}\).

\[d \vec{r}(q) = \vec{r}'(q) \, d q \ .\]

Arc-length parameter, \(s\). So that \(d s = |d \vec{r}(s)|\) and thus

\[|d \vec{r}(s)| = |\vec{r}'(s)| \, |d s| \qquad \rightarrow \qquad |\vec{r}'(s)| = 1 \qquad \rightarrow \qquad \vec{r}'(s) = \hat{t}(s) \ .\]

Frenet basis. Using arc-length parameter, Frenet basis is naturally defined as the set \(\{ \hat{t}, \hat{n}, \hat{b} \}\):

• tangent unit vector, \(\hat{t}(s) = \vec{r}'(s)\),

• normal unit vector, \(\hat{r}''(s) = \hat{t}'(s) =: \kappa(s) \, \hat{n}(s) \), with \(\kappa(s)\) local curvature

• binormal unit vector, \(\hat{b}(s) = \hat{t}(s) \times \hat{n}(s)\)

Using a general parameter, \(t\), with some abuse of notation \(\vec{r}(t) = \vec{r}(s(t))\) and indicating \(\dot{()} = \frac{d}{dt}\),

• \(\dot{\vec{r}} = \frac{d s}{d t} \frac{d \vec{r}}{d s} = \dot{s} \hat{t}\)

• \(\ddot{\vec{r}} = \dfrac{d}{dt} \dot{\vec{r}} = \dfrac{d}{dt} \left( \dot{s} \hat{t} \right) = \ddot{s} \hat{t} + \dfrac{ds}{dt} \dfrac{d}{ds} \hat{t} = \ddot{s} \hat{t} + \dot{s}^2 \kappa \, \hat{n}\)

Osculator circle. Circle with \(R(s) = \frac{1}{\kappa(s)}\), in plane orthogonal to \(\hat{b}(s)\), passing through \(\vec{r}(s)\), and thus center in \(\vec{r}_C(s) = \vec{r}(s) + \hat{n} R(s)\). Its parametric representation using its arc-length parameter \(p\), with \(\vec{r}(p=0) = \vec{r}(s)\) reads

\[\vec{r}(p) = \vec{r}_C(s) + R(s) \left[ - \cos \left(\frac{p}{R(s)} \right) \hat{n}(s) + \sin \left(\frac{p}{R(s)} \right)\hat{t}(s) \right] \ .\]

Its first and second order derivatives w.r.t. the arc-length \(p\) evaluated in \(p=0\), i.e. \(\vec{r} = \vec{r}(s)\) read:

• first derivative in \(p=0\),

  \[\left.\hat{t}(p)\right|_{p=0} = \left.\vec{r}'(p)\right|_{p=0} = \left. \left[ \sin \left(\frac{p}{R(s)} \right) \hat{n}(s) + \cos \left(\frac{p}{R(s)} \right)\hat{t}(s) \right] \right|_{p=0} = \hat{t}(s) \ ,\]

  i.e. the osculator circle has the same tangent as the curve in the point.

• second derivative in \(p=0\),

  \[\left. \kappa(p) \hat{n}(p)\right|_{p=0} = \left.\vec{r}''(p)\right|_{p=0} = \frac{1}{R(s)} \left. \left[ \cos \left(\frac{p}{R(s)} \right) \hat{n}(s) - \sin \left(\frac{p}{R(s)} \right)\hat{t}(s) \right] \right|_{p=0} = \frac{1}{R(s)}\hat{n}(s) = \kappa(s) \hat{n}(s) \ ,\]

  i.e. the osculator circle has the same normal vector and curvature as the curve in the point.

5.1.2. Surfaces

\[\vec{r}(q^1, q^2)\]

\[d \vec{r} = \frac{\partial \vec{r}}{\partial q^1} \, d q^1 + \frac{\partial \vec{r}}{\partial q^2} \, d q^2 = \vec{b}_1 \, d q^1 + \vec{b}_2 \, d q^2 \]

A third vector \(\vec{b}_3 := \hat{n}\) can be defined so that \(|\hat{n}| = 1\) and \(\hat{n} \cdot \vec{b}_{i} = 0\), \(i=1:2\). For \(i=1:2\), \(k=1:2\)

\[\frac{\partial \vec{b}_i}{\partial q^j} = \Gamma_{ij}^k \vec{b}_k = \Gamma_{ij}^1 \vec{b}_1 + \Gamma_{ij}^2 \vec{b}_2 + \Gamma_{ij}^3 \vec{b}_3\]

so that

\[\Gamma_{ij}^{k} = \vec{b}^k \cdot \frac{\partial \vec{b}_i}{\partial q^j}\]

Normal vector.

\[\vec{n}(q^1, q^2) = \frac{\partial \vec{r}}{\partial q^1}(q^1, q^2) \times \frac{\partial \vec{r}}{\partial q^2}(q^1, q^2) = \vec{b}_1(q^1, q^2) \times \vec{b}_2(q^1, q^2)\]

Tangent plane.

\[(\vec{r} - \vec{r}(q^1, q^2)) \cdot \vec{n}(q^1, q^2) = 0\]

Length of elementary segment.

\[\begin{split}\begin{aligned} |d \vec{r}|^2 & = d \vec{r} \cdot d \vec{r} = \\ & = \left( \vec{b}_1 \, d q^1 + \vec{b}_2 \, d q^2 \right) \cdot \left( \vec{b}_1 \, d q^1 + \vec{b}_2 \, d q^2 \right) = \\ & = g_{11} \, dq^1 \, dq^1 + g_{12} \, dq^1 \, dq^2 + g_{21} \, dq^2 \, dq^1 + g_{22} \, dq^2 \, d q^2 = g_{ij} \, dq^i \, dq^j \end{aligned}\end{split}\]

Second order approximation.

\[\begin{split}\begin{aligned} \vec{r}(q^1 + d q^1, q^2 + d q^2) & = \vec{r}(q_1, q_2) + \frac{\partial \vec{r}}{\partial q^i} \, dq^i + \frac{\partial^2 \vec{r}}{\partial q^i \partial q^j} \, dq^i \, dq^j = \\ & = \vec{r}(q_1, q_2) + \vec{b}_{i} \, dq^i + \vec{b}_k \Gamma^{k}_{ij} \, dq^i \, dq^j + \hat{n} \, \Gamma^{3}_{ij} \, dq^i \, dq^j \end{aligned}\end{split}\]

so that

\[\begin{split}\begin{aligned} \left[ \vec{r}(q^1 + d q^1, q^2 + d q^2) - \vec{r}(q^1, q^2) \right] \cdot \hat{n} & = \Gamma^3_{ij} \, d q^i \, dq^j = \\ & = \hat{n} \cdot \frac{\partial^2 \vec{r}}{\partial q^i \partial q^j} \, d q^i \, dq^j = \\ & = \hat{n} \cdot \frac{\partial^2 \vec{r}}{\partial q^i \partial q^j} \, \vec{b}^i \cdot \vec{b}_k d q^k \, \vec{b}^j \cdot \vec{b}_l dq^l = \\ & = \underbrace{d q^k \vec{b}_k}_{d \vec{r}} \cdot \left[ \hat{n} \cdot \frac{\partial^2 \vec{r}}{\partial q^i \partial q^j} \vec{b}^i \otimes \vec{b}^j \right] \cdot \underbrace{d q^l \vec{b}_l}_{d\vec{r}} \end{aligned}\end{split}\]

Curvature tensor.