Georg Reimann had a huge influence on mathematics and the way that we look at the mathematical world today. He started by taking mathematics courses from Moritz Stern and Gauss, learning from the best from the start of his mathematical career. In the spring of 1847, he moved to Berlin University to study under Steiner, Jacobi, Dirichlet, and Eisenstein. Focusing on elliptic function theory, Eisenstein and Riemann worked together to discuss theories and advances in the subject. But during his time at Berlin University, the man who influenced him the most was Dirichlet. This was the time that Riemann worked out his general theory of complex variables which stemmed off into some of his most important works. Riemann returned to Göttingen and began to work on his Ph.D. under Gauss, he officially finished and submitted his research in 1851. Riemann was influenced by many important mathematicians throughout his career, Weber and Listing were among some of the others that helped push him further into his research. Through working with these two men, Riemann gained a strong background in theoretical physics and important ideas of topology. He continued his work and started his thesis where he began researching the theory of complex variables and Riemann surfaces, he introduced topological methods into complex function theory, examined geometric properties of analytic functions, and completed research on conformal mappings and the connectivity of surfaces.
Riemann contributed many things to the mathematical world, but a very interesting part of his research consisted of Riemann spaces. While lecturing in Göttingen, he focused his lectures on geometry and defining a n-dimensional space as well as the definition of the curvature tensor. Now we ask the question, what is a Riemann space and why is it important? Before Riemann, there were not very many mathematicians that dove deep into the idea of n-dimensions but Riemann was determined to look deeper into what was really going on. He started thinking about geometry and how the geometry of space is not just one with straight lines or corners on a zero curvature plane. When we think of this statement, we typically think of Euclid and his development of geometry but Riemann wanted more from this. Shapes and planes more than two dimensional are more difficult for any person to picture, for example, can you picture what n-dimensions would look like? We see things like a table or a chair and just think, it has curves and the chair has a hole in the middle of it, but so what? We don’t think that a chair or a table can have positive or negative curvature like the other types of geometry, but we see things like this all the time. From this insight, Riemann developed a new geometrical concept called the Riemann space where force simply can be understood by geometry.
Riemann apparently borrowed from Gauss’s idea of a bookworm living on a two-dimensional piece of paper. Riemann took the idea that you place
these worms on a crumpled sheet of paper, what would this world look like from the worms two-dimensional view? What is interesting is that
the Riemann crumpled ball has all three types of
geometric curvature in a plane (sheet of paper), it contains zero curvature, positive curvature, and negative curvature. What do you think the world would look like to a book worm? According to Riemann, the world would look flat and undistorted because these creatures can only perceive two dimensions, they would not be able to detect that their bodies are crumpling with the paper in the higher dimensions as they move across their new world. Well why can’t the bookworm just move on top of the ball in a straight line? Riemann found that the “force” that would keep the bookworm from moving in a straight line is a result from the unseen warping from the third dimension (Cool, right?). Does this crumpled ball only have three dimensions? What if we were able to find more? Riemann didn’t stop with just the idea of the bookworm living in this new world, he cut the crumpled ball in half and found that on the inside, there are more dimensions to discover!
The approach to this new math opened new ways of mathematics, adding new dimensions to the way that we look at geometry and planes and thus proving why his research and discoveries were extremely important to our mathematical world.
Coming to the end of this post, you might still be asking why his discovery was so important? Well being in the field of mathematics and other fields like science, it is obvious that this discovery leads to the knowledge that other dimensions do exist in our world. It led to the discovery of math that was able to help calculate these dimensions and figure out just how many there were, which seems to be infinite! It was a milestone in math that helped develop where we are today, we are able to elaborate on the ideas of positive and negative curvature and go beyond just what we can see.