Through his pioneering contributions to differential geometry, Riemann laid the foundations of the mathematics of general relativity.

His father, Friedrich Bernhard Riemann, was a poor Lutheran pastor in Breselenz who fought in the Napoleonic Wars.

Riemann exhibited exceptional mathematical talent, such as calculation abilities, from an early age but suffered from timidity and a fear of speaking in public.

His teachers were amazed by his ability to perform complicated mathematical operations, in which he often outstripped his instructor's knowledge.

In 1846, at the age of 19, he started studying philology and Christian theology in order to become a pastor and help with his family's finances.

During the spring of 1846, his father, after gathering enough money, sent Riemann to the University of Göttingen, where he planned to study towards a degree in theology.

However, once there, he began studying mathematics under Carl Friedrich Gauss (specifically his lectures on the method of least squares).

Riemann held his first lectures in 1854, which founded the field of Riemannian geometry and thereby set the stage for Albert Einstein's general theory of relativity.

Riemann was a dedicated Christian, the son of a Protestant minister, and saw his life as a mathematician as another way to serve God.

[10] Riemann's published works opened up research areas combining analysis with geometry.

Over many months, Riemann developed his theory of higher dimensions and delivered his lecture at Göttingen on 10 June 1854, entitled Ueber die Hypothesen, welche der Geometrie zu Grunde liegen.

Riemann found the correct way to extend into n dimensions the differential geometry of surfaces, which Gauss himself proved in his theorema egregium.

For example, Riemann found that in four spatial dimensions, one needs ten numbers at each point to describe distances and curvatures on a manifold, no matter how distorted it is.

For the proof of the existence of functions on Riemann surfaces, he used a minimality condition, which he called the Dirichlet principle.

Through the work of David Hilbert in the Calculus of Variations, the Dirichlet principle was finally established.

When Riemann's work appeared, Weierstrass withdrew his paper from Crelle's Journal and did not publish it.

Weierstrass encouraged his student Hermann Amandus Schwarz to find alternatives to the Dirichlet principle in complex analysis, in which he was successful.

An anecdote from Arnold Sommerfeld[15] shows the difficulties which contemporary mathematicians had with Riemann's new ideas.

The physicist Hermann von Helmholtz assisted him in the work overnight and returned with the comment that it was "natural" and "very understandable".

By Ferdinand Georg Frobenius and Solomon Lefschetz the validity of this relation is equivalent with the embedding of

According to Detlef Laugwitz,[16] automorphic functions appeared for the first time in an essay about the Laplace equation on electrically charged cylinders.

Riemann gave an example of a Fourier series representing a continuous, almost nowhere-differentiable function, a case not covered by Dirichlet.

He also worked with hypergeometric differential equations in 1857 using complex analytical methods and presented the solutions through the behaviour of closed paths about singularities (described by the monodromy matrix).

The proof of the existence of such differential equations by previously known monodromy matrices is one of the Hilbert problems.

Through the summation of this approximation function over the non-trivial zeros on the line with real portion 1/2, he gave an exact, "explicit formula" for