We are at the 3rd topic for the event Modern Mathematics and I have learnt quite some interesting things so far with Topology Day and Chaos Theory Day, hopefully you did find them interesting. The next topic on the list is Differential Geometry. So let us get started:
Topology and Differential Geometry are quite close related. Differential geometry deals with metrical notions on manifolds, while differential topology deals with nonmetrical notions of manifolds. Explaining what a manifold is not not as straight forward as expected. A manifold is a topological space that is locally Euclidean. To illustrate this idea, consider the ancient belief that the Earth was flat as contrasted with the modern evidence that it is round. The discrepancy arises essentially from the fact that on the small scales that we see, the Earth does indeed look flat. In general, any object that is nearly “flat” on small scales is a manifold, and so manifolds constitute a generalization of objects we could live on in which we would encounter the round/flat Earth problem, as first codified by Poincaré.
Differential geometry arose and developed as a result of and in connection to the mathematical analysis of curves and surfaces. Mathematical analysis of curves and surfaces had been developed to answer some of the nagging and unanswered questions that appeared in Calculus, like the reasons for relationships between complex shapes and curves, series and analytic functions. These unanswered questions indicated greater, hidden relationships and symmetries in nature, which the standard methods of analysis could not address.
When curves, surfaces enclosed by curves, and points on curves were found to be quantitatively, and generally, related by mathematical forms the formal study of the nature of curves and surfaces became a field of study in its own right, with Monge’s paper in 1795, and especially, with Gauss’s publication of his article, titled ‘Disquisitiones Generales Circa Superficies Curvas’.
Initially applied to the Euclidean space, further explorations led to non-Euclidean space, and metric and topological spaces.
Differential Geometry has quite an impressive number of applications and I will mention just a couple of them bellow:
- in physics: one of the most important is Einstein’s general theory of relativity. According to the theory, the universe is a smooth manifold equipped with a pseudo-Riemannian metric, which describes the curvature of space-time. Understanding this curvature is essential for the positioning of satellites into orbit around the earth. Differential geometry is also indispensable in the study of gravitational lensing and black holes.
- in structural geology: used to analyze and describe geologic structures.
- in image processing and computer vision: used to process, analyse data on non-flat surfaces and analyse shapes in general.
Finally, here are a couple of books recommendations from introductory ones to ones which describe applications of differential geometry. Enjoy:
Main Source: Wikipedia;
There are many other sources for learning about differential geometry (especially because it has so many application in different other sciences) and I advice you to search and give it a try. At least a small glance over it and some of its wonderful concepts will make you see the world with different eyes.
Don’t forget that maths is everywhere! Enjoy! ~LThMath~