Author: Raymond O. Wells Jr
The geometry of manifolds and the perception of space
This essay discusses the development of key geometric ideas in the 19th century which led to the formulation of the concept of an abstract manifold (which was not necessarily tied to an ambient Euclidean space) by Hermann Weyl in 1913. This notion of manifold and the geometric ideas which could be formulated and utilized in such a setting (measuring a distance between points, curvature and other geometric concepts) was an essential ingredient in Einstein's gravitational theory of space-time from 1916 and has played important roles in numerous other theories of nature ever since.
fresh_42 said:I have found this paper on the internet and think it might be interesting for some on this forum because there are frequently questions similar to the ones the paper tries to answer.
http://arxiv.org/abs/1605.00890
http://arxiv.org/pdf/1605.00890v1.pdf
jedishrfu said:
A manifold is a mathematical concept that describes a space that is locally similar to Euclidean space, but may have a more complex global structure.
The geometry of manifolds is a branch of mathematics that studies the properties of geometric objects, such as curves and surfaces, in spaces that are not necessarily flat. This is in contrast to traditional geometry, which focuses on objects in Euclidean space.
The geometry of manifolds has many applications in physics, including in the study of general relativity, which describes the curvature of spacetime. It is also used in computer graphics to model and manipulate 3D objects.
The geometry of manifolds is closely related to other fields of mathematics, such as topology, differential geometry, and algebraic geometry. It also has connections to physics, specifically in the areas of quantum mechanics and string theory.
Current research in the geometry of manifolds includes the study of different types of manifolds, such as Riemannian manifolds and symplectic manifolds, and their properties. There is also ongoing research in the application of manifolds to other areas, such as machine learning and data analysis.