1. 1D, 2D, 3D…. nD Euclidian, Spherical, Hyperbolic General Riemannian
SPACE
1D, 2D, 3D…. nD
Euclidian, Spherical, Hyperbolic
General Riemannian
MATERIALS SCIENCE
&
ENGINEERING
Anandh Subramaniam & Kantesh Balani
Materials Science and Engineering (MSE)
Indian Institute of Technology, Kanpur
URL: home.iitk.ac.in/~anandh
AN INTRODUCTORY E-BOOK
Part of
A Learner’s Guide

2. SPACE
A simplistic point of view is that space is a perfect inert vacuum and matter resides in that space.
The real picture is more complicated, with actual space deviating from the ‘smooth picture’ at the very small lengthscales.  Space at these lengthscales is teeming with virtual particles which fleetingly come into existence.
There are unsolved questions regarding the number of dimensions and curvature of space we live in → though it is assumed to be 3D Euclidean locally* → At the scale of the universe it is understood to be non-Euclidean
Many of the theories of physics which describe nature (e.g. the string theories) require higher dimensions (10 or more in some of them!).  Dimensions higher than 3 are supposed to be ‘compactified’
Apart from the possibility of higher spatial dimensions, in Einstein’s description of gravity, the 3D space is intertwined with time into a 4D ‘space-time unit’.
In some theories, the structure of glasses and quasicrystals are described by hyper-dimensional constructs.
* refer next slide

3. Sum of the Included angles of a triangle
SPACE continued…
Gaussian curvature is the product of two orthogonal curvatures
Mean curvature is the average of two orthogonal curvatures
Space can be:  Euclidean (flat) → Zero (0) Gaussian Curvature  Spherical → Positive (+) Gaussian Curvature  Hyperbolic → Negative () Gaussian Curvature
If for a Hyperbolic surface the mean curvature is zero then the surface is called a Minimal Surface
Space
Gaussian Curvature
Sum of the Included angles of a triangle
Euclidean
180
Spherical +
> 180
Hyperbolic 
< 180
Emphasis though it is assumed that space is:  3D → in general it can be nD  Euclidean → it can be Non-Euclidean with local variations in the curvature of space