Differential geometry (DG): calculus applied to curved geometry.

Non-Euclidean geometries: spherical and hyperbolic

Today, non-Euclidean means hyperbolic

Spherical: there are non parallel lines
Euclidean: there is 1 parallel line to a given line (can't be verified by experiment)
Hyperbolic: there are two parallel lines

Angular excess = ϵ = (angle sum of triangle) - π

Spherical: ϵ > 0
Euclidean: ϵ = 0 (experimentally verifiable)
Hyperbolic: ϵ < 0

For a sphere of radius R, the relation between ϵ and the area of the triangle, A:

A = ϵR² => ϵ = A/R²

In general, ϵ = κA, with κ = curvature (constant, positive in spherical and negative in hyperbolic).

Shortest path between two points: geodesic. But two problems:

1) in a sphere, two geodesics connect points a and b, one is the shortest, the other is not
2) poles are connected by multiple geodesics

Better to consider short geodesic segments: local characterisation of the minimisation of length.
Short geodesic segments can be extended, and that extension is unique.
The length of the short geodesic segment is what we call distance between two close points.

Geodesic segments provide: shortest and straightest routes.

Geodesic circle: pick a point and draw all points at distance r.
Its length will be smaller than 2πr.

Given three points we can build a geodesic triangle if we connect each pair of vertices with geodesic segments.

Intrinsic geometry: what is knowledgeable to an ant on a given surface.
The ant defines a geodesic distance, and then it can build triangles and so on.
If the surface is bent, without straining or tearing, nothing changes for the ant.
In a bent paper, instrinsic geometry is still Euclidean.

Extrinsic geometry changes with bending.

Experiment 1:
- mark two points on a vegetable or fruit
- connect them with a stretched string
- draw the two sides of the string
- remove the string and peel the strip
- unfold the strip, which will be flat and straight
- the (narrow) strip cannot bend within the tangent plane, only perpendicular to it
- the stretched string only works on convex parts

Experiment 2:
- use a narrow tape, mark a point and start sticking the tape along a direction
- that will be a geodesic
- it will not bring you to a desired destination point
- this method works on concave and conves parts

Consider an ant on a corner of a 8×8×10 room. What is the shortest path to the opposite corner of the room? Easy:
- unfold the room, as if made of paper
- the opposite corner will appear more than once (three times) in the unfolded structure
- test the shortest straight line, and that will be the desired path
- it will be sqrt(356) ≈ 18.867
- the ant doesn't care about the extrinsic angles. For her, it's a plane.

Since the sum of the angles in a triangle must be positive, in hyperbolic geometry we have the threshold ϵ >= -π => π/|κ| is the maximum angle of a triangle.

If ϵ ≠ 0, no similar triangles (of different size) exist!

If ϵ ≠ 0, there is an absolute unit of length! For example in spherical geometry one could define the unit as the side of THE equilateral angle. Another choice: a length R such that κ = 1/R² (spherical, where R is the radius), or κ = -1/R².

If a triangle is very small, it will seem flat.