Geometry of space, physics and analysis* by Shing-Tung Yau† We must admit with humility that, while number is purely and Chinese discovered the concept of differentiation a product of our minds, space has a reality outside our minds, of trigonometric functions. so that we cannot completely prescribe its properties a priori. Copernicus proposed that the sun, instead of the C. F. Gauss (Letter to Bessel, 1830) earth, is the center of the universe. The work of Kepler The concept of space has gone through many and Galileo that supported Copernicanism appeared stages of evolution. Many of them are related to the a half century later. It was widely accepted only after advancement of our understanding of nature. Newton formulated the universal law of gravitation. When Isaac Newton merged the development of differential and integral calculus together and veri- From Ancient to Modern Geometry fied Kepler’s laws of motion of planets, mathematics reached a new plateau, largely because calculus gave In the days of the Greek geometers, plane and us a most powerful tool to understand and calculate Euclidean geometry were reasonably adequate to de- the geometry of curved objects that appear in nature. scribe most observations. The Greek scholars were Some historians said that Leibniz independently convinced that the earth is round and were able to discovered calculus. Many of the mathematical nota- measure the diameter of the earth and its distance to tions we used today are in fact due to him. However, the sun based on plane geometry. The Chinese schol- without Newton’s important application of calculus ars also measured the distance of the earth to the to mechanics, few people would have paid much at- heaven using similar ideas. tention to the development of calculus. But even then, Archimedes had already started to Soon afterwards, many great mathematicians de- investigate infinite processes to understand geomet- veloped new disciplines of mathematics based on this ric figures that could not be described by Euclidean new tool of calculus. The most notable was the devel- geometry alone. Calculations of volumes of many dif- opment of differential geometry and the introduction ferent geometric figures started the investigation of of calculus of variations. Fermat, Euler and Monge are integral calculus, as was also proposed by ancient among the most notable contributors. Some of their Chinese mathematicians. works are still being investigated today. It is amazing Not long afterwards, in the process of measuring that even then, calculus of variations was used exten- the movement of celestial bodies, the ancient Indians sively by differential geometers. In his famous paper On the Hypotheses Which * This article is based on the author’s talk delivered at Ts- inghua University in the summer of 2015, on the hundredth Lie at the Foundation of Geometry (1854), Riemann anniversary of the discovery of general relativity. The au- wrote: “The theorems of geometry cannot be deduced thor’s works on geometry and physics are collected in a from the general notion of magnitude alone, but two-volume set, Selected Expository Works of Shing-Tung Yau only from those properties which distinguished space with Commentary, International Press, 2014. from other conceivable entities, and these properties † Department of Mathematics, Harvard University, Cam- bridge, MA 02138, USA can only be found experimentally … This takes us E-mail: [email protected] into the realm of another science – physics.” JULY 2016 NOTICES OF THE ICCM 1 The discovery of the concept of the intrinsic Geometry and Topology of Manifolds Gauss curvature is certainly a landmark for the mod- While Newton proposed the space to be static, ern development of differential geometry. It should Einstein demanded the space to be dynamic, integrat- have been a major inspiration for the birth of Rieman- ing special relativity with Newtonian gravity. The idea nian geometry. of Minkowski to interpret special relativity using ge- In the beginning of his famous address in Göttin- ometry of four-dimensional spacetime was essential gen in 1854, Riemann says that the foundations had for Einstein’s development of general relativity. This already been laid by Gauss’ investigations of curved led Einstein and Grossmann to use the work of Rie- surfaces. A couple of years later, when he introduced mann and Ricci. the concept of Riemannian curvature tensor in an es- When Einstein developed the theory of general say for a competition given by the French Academy of relativity, he consulted, besides Grossmann, Levi- Sciences, he mentioned that the calculation was sim- Civita and David Hilbert. It was clear that Hilbert had ilar to the one given by Gauss. given tremendous help to Einstein and Hilbert had It should be noted that Riemann was interested in even derived the action for general relativity before his geometry because of his strong desire to find an Einstein. adequate concept of space to understand the laws of After the theory of general relativity was accom- nature. He was interested in the laws of electromag- plished, a group of scientists were interested in gen- netic field. In his collected works published posthu- mously in 1876, he discusses electrodynamics and eralizing the theory to incorporate other fields. Levi- apparently anticipates some of Maxwell’s work. Rie- Civita was the one who developed the concept of dif- mann had originally submitted the paper in 1858, but ferentiation on a curved manifold, which he called then withdrew it because of an error in a calculation. the theory of connections. In order for the connection Maxwell independently published his equations be- to behave like differentiation in Euclidean space, he tween 1861 and 1862. demands the connection to be symmetric. If parallel In his famous address, Riemann asked how to transportation preserves the metric, the connection come up with a concept of geometry that can incorpo- would be uniquely determined by the metric. Such a rate the space of the enormously large and the space connection is called Levi-Civita connection. of the extremely small. He even raised the possibility Levi-Civita observed that if the connection is not that a discrete space can play an important role. assumed to be symmetric, the skew-symmetric part It is a curiosity whether Riemann would have dis- of the connection can be used as a free variable to be covered general relativity if he had lived forty years a new field to be considered in physics. This inspired longer. (He died on July 20, 1866, less than forty years both Élie Cartan and Hermann Weyl to develop the old.) In any case, Einstein’s theory of general relativ- theory much further. ity was based on Riemannian geometry, as was in- Weyl studied connections on bundles and hence troduced to him by Grossmann in 1910. General rel- initiated the fundamental concept of gauge theory ativity is one of the most spectacular achievement with abelian gauge groups, and this was later gen- of physics in the twentieth century. I shall comment eralized by Yang and Mills in 1954 to include non- later on how the theory of general relativity influ- abelian gauge groups. But it was Weyl who under- enced the modern development of differential geom- stood Maxwell’s equations as a gauge theory and etry. hence gave topological meaning to the theory of elec- At the same time when Riemann introduced his tricity and magnetism. Later in 1970, ’t Hooft, us- geometry, he was developing the geometric aspect ing ideas of DeWitt and Faddeev-Popov, succeeded of complex analysis and in particular he studied the in quantizing Yang-Mills gauge theory. This then led concept of a Riemann surface – one of the most fun- to the current fundamental understanding of particle damental tool in modern mathematics and physics. physics based on nonabelian gauge theory in which Not only did he give deep insights into complex anal- the Standard Model was built. ysis from the point of view of geometry, but also All these spectacular works in modern physics he initiated the full development of modern topol- are related to parallel developments from the strong ogy. He studied the connectivity of a space and in- desire of topologists and geometers to understand troduced several measurements of high connectiv- the structure of manifolds. The basic idea is to sim- ity. They were later called Betti numbers of those plify complicated spaces by cutting them into simple spaces. building blocks. The concept of conformal geometry and confor- The simplest building blocks are cells, con- mal invariance have been major tools for string the- tractible spaces or handle bodies. In the process of ory. And its higher dimensional generalizations to attaching simple spaces together, we often need to complex geometry have been fundamental in under- twist before gluing them together. Hence, this led to standing complex manifolds. the notion of a fiber space. 2 NOTICES OF THE ICCM VOLUME 4,NUMBER 1 The concept of a fiber bundle was introduced in terior differentiation. This is a concept which general- early days, first by Whitney in 1935 for sphere bun- izes the divergence and curl in multivariable calculus. dles over any spaces. (Seifert considered 3-manifolds In the early twentieth century, Poincaré, Cartan which can be written as circle bundles in 1932, but and de Rham uncovered the topological meaning of the point of view is different.) The theory was stud- the differential forms and their exterior derivatives, ied by Stiefel, Hopf, Ehresmann, Pontryagin, Chern, which inspired Hodge and Kodaira to develop the the- Steenrod, Leray and Serre. ory of harmonic forms; some of their ideas can be The characteristic classes introduced by the traced back to the classical theory of fluid dynamics founders of the theory of fiber bundles had deep in- and electricity and magnetism.