微分几何

主要参考 Visual Differential Geometry and Forms (A mathematical drama in five acts)

The Nature of Space

Euclidean and Non-Euclidean Geometry

• (Euclidean Geometry) Parallel Axiom:

  Through any point $p$ not on the line $L$ there exists precisely one line $P$ that is parallel to $L$.

• (Hyperbolic Geometry) Hyperbolic Axiom:

  There are at least two parallel lines through $p$ that do not meet $L$.

• (Spherical Geometry) Spherical Axiom:

  There are no lines through $p$ that are parallel to $L$: every line meets $L$.

• Angular Excess: $\mathcal{E}=$ (angular sum of triangle) $-\pi$

• Intrinsic v.s. Extrinsic geometry

• Geodesic: (1) the shortest route between two points that are not too far apart; (2) the "straightest" route.

  To construct a geodesic on a surface, emanating from a point $p$ in direction $\overrightarrow{v}$, stick one end of a length of narrow sticky tape down at $p$ and unroll it onto the surface, starting in the direction $\overrightarrow{v}$.

• $$\mathcal{E}(\mathrm{\Delta })=\mathcal{K}\mathcal{A}(\mathrm{\Delta })$$

  $\mathcal{K}=+(1/R^2)$ in Spherical Geometry;

  $\mathcal{K}=-(1/R^2)$ in Hyperbolic Geometry.

Gaussian Curvature

• $$\mathcal{K}(p)=\underset{{\mathrm{\Delta }}_p\to p}{lim}\frac{\mathcal{E}({\mathrm{\Delta }}_p)}{\mathcal{A}({\mathrm{\Delta }}_p)}$$

• $$\mathcal{K}\asymp \frac{3}{\pi }\left[\frac{2\pi r-\mathcal{C}(r)}{r^3}\right]$$$$\mathcal{K}\asymp \frac{12}{\pi }\left[\frac{\pi r^2-\mathcal{A}(r)}{r^4}\right]$$

• pseudospherical (constant negative curvature)

  pseudosphere

  

• Local Gauss-Bonnet Theorem:

  $$\mathcal{E}(\mathrm{\Delta })=\alpha +\beta +\gamma -\pi ={\iint }_{\mathrm{\Delta }}\mathcal{K}\mathrm{d}\mathcal{A}$$

  

The Metric

Mapping Surfaces: The Metric

• Metric: the rule giving $\delta \hat{s}$ in terms of $\delta z$$$\mathrm{d}\hat{s}=\mathrm{\Lambda }(z,\gamma )\mathrm{d}s$$

• The general metric formula for a general surface is

  $$\mathrm{d}{\hat{s}}^2={\mathcal{A}}^2\mathrm{d}{u}^2+{\mathcal{B}}^2\mathrm{d}{v}^2+2\mathcal{A}\mathcal{B}\cos \omega \mathrm{d}u\mathrm{d}v$$$$\mathcal{A}\equiv \left(\frac{\partial {\hat{s}}_1}{\partial u}\right),\mathcal{B}\equiv \left(\frac{\partial {\hat{s}}_2}{\partial v}\right)$$
• $$\mathcal{K}=-\frac{1}{\mathcal{A}\mathcal{B}}({\partial }_v\left[\frac{{\partial }_v\mathcal{A}}{\mathcal{B}}\right]+{\partial }_u\left[\frac{{\partial }_u\mathcal{B}}{\mathcal{A}}\right])$$

• Conformal: preserves the magnitude and sense (anticlockwise or clockwise) of angles

  Anticonformal: preserves the magnitude but reverses the sense

• Conformal map $\iff \mathrm{d}\hat{s}=\mathrm{\Lambda }(z)\mathrm{d}s$

• Conformal / Isothermal coordinates: $\mathcal{A}=\mathcal{B}=\mathrm{\Lambda }$