DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES 7. Geodesics and the Theorem of Gauss-Bonnet

Let’s understand first what means “straight” line in the plane. If you want to go from a point in a plane “straight” to another one, your trajectory will be such a line. In other words, a straight line L has the property that if we fix two points P and Q on it, then the piece of L between P and Q is the shortest curve in the plane which joins the two points. Now, if instead of a plane (“flat” surface) you need to go from P to Q in a land with hills and valleys (arbitrary surface), which path will you take? This is how the notion of geodesic line arises: “Definition” 7.1.1. Let S be a surface. A curve α : I → S parametrized by arc length is called a geodesic if for any two points P = α(s1), Q = α(s2) on the curve which are sufficiently close to each other, the piece of the trace of α between P and Q is the shortest of all curves in S which join P and Q.