Riemann’s Novel Lecture that Changed the Course of Geometry

In 1915, Albert Einstein dramatically altered our perception of gravity with his theory of general relativity, primarily emphasizing the way mass and energy distort the four-dimensional fabric of spacetime. The essential geometric framework for this theory can be traced back to mathematician Georg Friedrich Bernhard Riemann, who developed a unique form of geometry—elliptic geometry—that extends beyond the confines of traditional Euclidean geometry.

Bernhard Riemann (1826–1866) was a German mathematician associated with the University of Göttingen. At Göttingen, he was mentored by the illustrious Carl Friedrich Gauss (1777–1855), whose extensive contributions to mathematics are well-known. Riemann also spent time at the University of Berlin, where he learned from notable figures such as Jacobi, Dirichlet, and Eisenstein, with Dirichlet having a particularly significant influence on his work.

> According to Felix Klein: > > “Riemann was bound to Dirichlet by the strong inner sympathy of a like mode of thought. Dirichlet loved to make things clear to himself in an intuitive substrate; along with this he would give acute, logical analyses of foundational questions and would avoid long computations as much as possible. His manner suited Riemann, who adopted it and worked according to Dirichlet’s methods.”

Gauss praised Riemann's "gloriously fertile originality" after overseeing his PhD thesis in complex variables. Years before Riemann's time, Gauss had begun questioning the long-standing principles of Euclidean geometry, as laid out in Elements, which only addressed two-dimensional and three-dimensional constructs without considering higher dimensions. He encouraged Riemann to explore reformulating Euclidean geometry to include curved surfaces.

During this period, the notion of a fourth dimension was often dismissed as absurd. John Wallis referred to it in his Treatise of Algebra (1685) as a “Monster in Nature, and less possible than a Chimera, or Centaure.” Meanwhile, János Bolyai and Nikolai Lobachevsky were pioneering hyperbolic geometry, which provided a new avenue for Riemann's explorations.

Riemann’s Breakthrough Lecture

To secure a permanent position at the University of Göttingen, Gauss recommended Riemann for the Habilitation qualification, which required an extensive dissertation and a lecture. Riemann dedicated 2.5 years to his dissertation on representing functions through trigonometric series.

In 1854, he presented a groundbreaking lecture titled “On the Hypotheses Which Lie at the Foundations of Geometry,” which laid the foundation for what is now known as Riemannian geometry. The lecture began with definitions of n-dimensional space, geodesics, and curvature tensors, ultimately connecting these concepts to the tangible world.

Many attendees struggled to comprehend Riemann's ideas, as they were far ahead of their time. However, Gauss fully understood and appreciated the relevance of Riemann's geometry to curved surfaces.

M. Monastyrsky notes: > “Among Riemann’s audience, only Gauss was able to appreciate the depth of Riemann’s thoughts. … The lecture exceeded all his expectations and greatly surprised him. Returning to the faculty meeting, he spoke with the greatest praise and rare enthusiasm to Wilhelm Weber about the depth of the thoughts that Riemann had presented.”

In Hyperspace, Michio Kaku states: > “In retrospect, this was, without question, one of the most important public lectures in the history of mathematics. Word spread quickly throughout Europe that Riemann had decisively broken out of the confines of Euclidean geometry that had ruled mathematics for 2 millennia. News of the lecture soon spread throughout all the centers of learning in Europe. His talk was translated into several languages and created quite a sensation in mathematics. There was no turning back to the work of Euclid.”

Comparison between Euclidean and Riemannian Geometry

• Euclidean geometry pertains to flat spaces such as points, lines, and planes, while Riemannian geometry applies to curved spaces like cylinders, spheres, and tori.
• The fifth postulate of Euclid, known as the parallel postulate, is entirely dismissed in elliptic geometry, which states, “Through a point not on a given line, there is only one line parallel to the given line,” whereas in curved geometry, there are no parallel lines.
• In flat spaces, the sum of the angles in a triangle always equals 180°, while in curved spaces, the sum can be greater or less than 180° due to the bending of triangle sides.
• The shortest distance between two points in flat spaces is a straight line, calculable via a distance formula, while in curved spaces, straight lines are termed geodesics, which can sometimes present multiple paths between two points.

Gauss described that a vector in one dimension has two components—magnitude and direction. Riemann expanded this concept to higher dimensions, showing that a vector in three dimensions could have six independent components to describe curvature, while in four dimensions, this number rises to twenty.

> “A tensor is a vector on steroids.” — Walter Isaacson.

How This Outstanding Work Helped Einstein in Generalizing His Theory of Gravity

When developing his special theory of relativity, Einstein primarily focused on its physical interpretations rather than its mathematical underpinnings. His former teacher, Hermann Minkowski, later established the geometry of special relativity, merging space and time into a single concept.

For his general theory, however, Einstein was unaware of the mathematical principles governing gravitational effects around massive objects.

> “Recently, I have been working furiously on the gravitation problem. It has now reached a stage in which I am ready with the statics. I know nothing as yet about the dynamic field, that must follow next. … Every step is devilishly difficult.” > > — (letter to M. Besso, March 26, 1912)

Convinced that the description of gravity via a scalar field needed re-evaluation, he sought assistance from his mathematician friend Marcel Grossmann at Zurich Polytechnic. Grossmann urged him to explore Riemann's new geometric framework.

This mathematical approach proved pivotal for Einstein, leading him to conclude that gravity arises from spacetime curvature—the more pronounced the curvature, the stronger the gravitational force.

As summarized by Misner, Thorne, and Wheeler: > “Matter tells spacetime how to curve, and spacetime tells matter how to move.”

Additionally, Einstein utilized the generally covariant formalism and tensor calculus developed by Italian mathematicians Gregorio Ricci and Levi-Civita, rooted in the works of Gauss, Riemann, and Christoffel.

Once he recognized Riemannian geometry as the suitable mathematical tool for his theory, the following three years became a period of intense effort in his research. “I am exhausted. But the success is glorious,” Einstein remarked in 1915 after completing a decade-long endeavor.

Einstein acknowledged Riemann’s contributions in profound terms: > “Physicists were still far removed from such a way of thinking: space was still, for them, a rigid, homogeneous something, susceptible of no change or conditions. Only the genius of Riemann, solitary and uncomprehended, had already won its way by the middle of the last century to a new conception of space, in which space was deprived of its rigidity, and in which its power to take part in physical events was recognized as possible.”

Hans Freudenthal noted: > “The general theory of relativity splendidly justified his work. In the mathematical apparatus developed from Riemann’s address, Einstein found the frame to fit his physical ideas, his cosmology, and cosmogony: and the spirit of Riemann’s address was just what physics needed: the metric structure determined by data.”

Riemann made significant contributions to various fields including number theory, analysis, functions, and the topological properties of curved surfaces. Unfortunately, after suffering from tuberculosis due to a severe cold in 1862, he passed away at the young age of 39 in 1866 at Selasca, Italy.

Dedekind noted: > “His strength declined rapidly, and he himself felt that his end was near. But still, the day before his death, resting under a fig tree, his soul filled with joy at the glorious landscape, he worked on his final work which unfortunately, was left unfinished.”

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