Differential Geometry and Topology of Curves 2024, 35: Curves and Vector Functions

I've never heard differential geometry, because I'm afraid of geometry. Nevertheless, I've recently run out of math topics I wanted to check out, and I've heard it's good to know. I thought I'd start with something that also has a intersection with topology, just so it at least feels a little familiar. As such, I took "Differential Geometry and Topology of Curves" by Yu Animov. It starts off with the notion of curves, which apparently are important in differential geometry as well. By the various ways of defining curves, an elementary curve is a set of the Euclidean space which is a set of an image of an interval of the real axis under a homeomorphic map. An elementary curve can be decomposed into continuous functions of its coordinates, and further parametrized through monotone continuous functions. It's simple, if it's an image of a segment of the real axis (or topologically equivalent) under a homeomorphism. The image of a circle under a homeomorphism is a closed Jordan curve. A parameter t varying an on an interval (a, b) can be assigned a vector-valued function r(t). If the coordinates are Cartesian, r(t) is equivalent to the representation of three scalar-valued functions. r(t) can be treated as a regular function, along with its potential limits. Continuity is functionally defined the same way as it is for scalar functions, which gives rise to the definition of vector-function derivatives. For integrals however, new properties exist for vector-valued functions. For some constant vector a,

A curve is considered Cᵏ-regular, iff there is a parametrization so that each component of the position vector r(t) is a Cᵏ-regular function without vanishing. For k = 1, the curve is considered smooth. A one-to-one projected segment [a, b] onto a curve implies one of the simplest representations. r = {x, y(x), z(x)}. The position vector of γ parametrized by x is of this form. Let Φ(x, y) be a C¹-regular function. If a point P with fixed points satisfies Φ(x, y) = 0 with grad Φ ≠ 0 at P, then points around a small neighbourhood of P form a C¹-regular curve. A point P of a curve and a set in some neighbourhood U of it, so that P decomposes U into 2 half-neighbourhoods, and another point of the same curve Q. If Q is situated in one half of U, tending to P, then the limit position of the ray PQ originating in P is the "ray tangent" to the curve at P wrt. said half of U. If both rays tangent to it at P, and they form a straight line, then it's a "straight line tangent". For any point P of a smooth curve there exists a tangent at P and a directing vector of the tangent. Vectors orthogonal to the tangents are "normal" to the curve.