For the second part of the lecture, Riemann discussed the dimension of real space and what geometry one should use to describe it. This laid the foundation for the field of Riemannian geometry. First, the question of how we might define an n-dimensional space resulted in the definition of Riemann space, including the Riemann tensor. It wasn’t published until two years after his death, in 1866, and is now considered one of the most important works in geometry. He used the opportunity to develop a highly original theory of higher dimensions, described in a lecture –“On the Hypotheses Which Lie at the Foundations of Geometry”–delivered on June 10, 1854, that included a workable definition of how one might measure the curvature of space. Riemann did not disappoint his mentor, despite a phobia of public speaking. Gauss described Riemann as having “a gloriously fertile originality” in his report on the thesis, and two years later, when Riemann was required to give a lecture to land a faculty position at Göttingen, Gauss assigned his star pupil the topic of the foundations of geometry–a seemingly mundane topic in the hands of a lesser mathematician.
Two years later, in 1849, he returned to Göttingen to pursue his PhD with Gauss, completing his thesis in 1851 on the theory of complex variables, the basis for what we now call Riemann surfaces. With his parents’ blessing, Riemann transferred to the University of Berlin the following year, studying under some of the most prominent mathematicians of his time. But then he attended lectures by Carl Friedrich Gauss and Moritz Stern, who inspired him to switch his studies. In 1846, his father scraped together sufficient funds to send his son to the University of Göttingen, where Riemann initially intended to study theology so that he could help support his family. The young Riemann was shy and nervous, but gifted in mathematics–so much so that while attending high school in Hannover, his knowledge sometimes surpassed that of his teachers. The geometric foundation for his work was laid some 60 years earlier, with the work of a German mathematician named Georg Friedrich Bernhard Riemann.īorn in what is now the Federal Republic of Germany in 1826, Riemann was the second of six children of a Lutheran pastor, who taught his son until he turned ten. Albert Einstein changed our view of the universe in 1915 when he published the general theory of relativity, in which he set forth the notion of a four-dimensional spacetime that warps and curves in response to mass or energy.
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