Georg Friedrich Bernhard Riemann (1826 – 1866) was an extremely influential German mathematician who made important contributions to analysis and differential geometry, some of them paving the way for the later development of general relativity.
Riemann was the second of six children. The family was not well off and suffered from poverty and under nourishment. Riemann was shy, terrified of speaking in public and during his life suffered from numerous nervous breakdowns. He also suffered from tuberculosis and eventually died of tuberculosis on his third journey to Italy in 1866 aged only 40.
In 1846, at the age of 19, he started studying philology and theology in order to become a priest and help with his family's finances, but in 1847, his father managed to raise enough money to send Riemann to university, allowed him to stop studying theology and start studying mathematics.
He was sent to the University of Göttingen, where he studied under Carl Friedrich Gauss. In 1853, Gauss asked his student Riemann to prepare a paper on the foundations of geometry. Riemann spent many months developing his theory, struggling with the effects of ill health. He suffered a nervous breakdown in 1854. When he finally delivered his lecture at Göttingen in 1854, the mathematicians present received it with enthusiasm.
Later recognised to be one of the most important works in geometry, it was titled Über die Hypothesen welche der Geometrie zu Grunde liegen (loosely: "On the foundations of geometry"; more precisely, "On the hypotheses which underlie geometry"), and was published in 1868.
The subject founded by this work is Riemannian geometry. Riemann found the correct way to extend into n dimensions, the differential geometry of surfaces, which Gauss himself proved in his theorema egregium. The fundamental object is called the Riemann curvature tensor.
The lecture not only served to provide the foundation for the field of Riemannian geometry but set the stage for Einstein's general relativity. Within 3 decades of the talk, Einstein had used Riemann geometry to explain the nature of ‘four dimensional’ space.
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