In the

mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

field of

differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...

, there are various splitting theorems on when a

pseudo-Riemannian manifold In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...

can be given as a metric product. The best-known is the Cheeger–Gromoll splitting theorem for Riemannian manifolds, although there has also been research into splitting of

Lorentzian manifold In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the r ...

Cheeger and Gromoll's Riemannian splitting theorem

Any connected

Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...

has an underlying

metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...

structure, and this allows the definition of a ''geodesic line'' as a map such that the distance from to equals for arbitrary and . This is to say that the restriction of to any bounded interval is a curve of minimal length which connects its endpoints. In 1971,

Jeff Cheeger Jeff Cheeger (born December 1, 1943, Brooklyn, New York City) is a mathematician. Cheeger is professor at the Courant Institute of Mathematical Sciences at New York University in New York City. His main interests are differential geometry and ...

and

Detlef Gromoll Detlef Gromoll (13 May 1938 – 31 May 2008) was a mathematician who worked in Differential geometry. Biography Gromoll was born in Berlin in 1938, and was a classically trained violinist. After living and attending school in Rosdorf and gra ...

proved that, if a

geodesically complete In mathematics, a complete manifold (or geodesically complete manifold) is a (pseudo-) Riemannian manifold for which, starting at any point , you can follow a "straight" line indefinitely along any direction. More formally, the exponential map a ...

and connected Riemannian manifold of nonnegative

Ricci curvature In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure ...

contains any geodesic line, then it must split isometrically as the product of a complete Riemannian manifold with . The proof was later simplified by Jost Eschenburg and Ernst Heintze. In 1936,

Stefan Cohn-Vossen Stefan Cohn-Vossen (28 May 1902 – 25 June 1936) was a mathematician, who was responsible for Cohn-Vossen's inequality and the Cohn-Vossen transformation is also named for him. He proved the first version of the splitting theorem. He was also ...

had originally formulated and proved the theorem in the case of two-dimensional manifolds, and Victor Toponogov had extended Cohn-Vossen's work to higher dimensions, under the special condition of nonnegative

sectional curvature In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature ''K''(σ''p'') depends on a two-dimensional linear subspace σ''p'' of the tangent space at a poi ...

. The proof can be summarized as follows. The condition of a geodesic line allows for two

Busemann function In geometric topology, Busemann functions are used to study the large-scale geometry of geodesics in Hadamard spaces and in particular Hadamard manifolds (simply connected complete Riemannian manifolds of nonpositive curvature). They are named aft ...

s to be defined. These can be thought of as a normalized Riemannian distance function to the two endpoints of the line. From the fundamental ''Laplacian comparison theorem'' proved earlier by

Eugenio Calabi Eugenio Calabi (born 11 May 1923) is an Italian-born American mathematician and the Thomas A. Scott Professor of Mathematics, Emeritus, at the University of Pennsylvania, specializing in differential geometry, partial differential equations and ...

, these functions are both

superharmonic An overtone is any resonant frequency above the fundamental frequency of a sound. (An overtone may or may not be a harmonic) In other words, overtones are all pitches higher than the lowest pitch within an individual sound; the fundamental i ...

under the Ricci curvature assumption. Either of these functions could be negative at some points, but the

triangle inequality In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but ...

implies that their sum is nonnegative. The

strong maximum principle In the mathematical fields of partial differential equations and geometric analysis, the maximum principle is any of a collection of results and techniques of fundamental importance in the study of elliptic and parabolic differential equations. ...

implies that the sum is identically zero and hence that each Busemann function is in fact (weakly) a

harmonic function In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is, : \f ...

. Weyl's lemma implies the infinite differentiability of the Busemann functions. Then, the proof can be finished by using

Bochner's formula In mathematics, Bochner's formula is a statement relating harmonic functions on a Riemannian manifold (M, g) to the Ricci curvature. The formula is named after the American mathematician Salomon Bochner. Formal statement If u \colon M \righ ...

to construct parallel vector fields, setting up the de Rham decomposition theorem. Alternatively, the theory of

Riemannian submersion In differential geometry, a branch of mathematics, a Riemannian submersion is a submersion from one Riemannian manifold to another that respects the metrics, meaning that it is an orthogonal projection on tangent spaces. Formal definition Let (' ...

s may be invoked. As a consequence of their splitting theorem, Cheeger and Gromoll were able to prove that the

universal cover A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete spa ...

of any

closed manifold In mathematics, a closed manifold is a manifold without boundary that is compact. In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components. Examples The only connected one-dimensional example ...

of nonnegative Ricci curvature must split isometrically as the product of a closed manifold with a

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...

. If the universal cover is topologically

contractible In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within that ...

, then it follows that all metrics involved must be

flat Flat or flats may refer to: Architecture * Flat (housing), an apartment in the United Kingdom, Ireland, Australia and other Commonwealth countries Arts and entertainment * Flat (music), a symbol () which denotes a lower pitch * Flat (soldier), ...

Lorentzian splitting theorem

In 1982,

Shing-Tung Yau Shing-Tung Yau (; ; born April 4, 1949) is a Chinese-American mathematician and the William Caspar Graustein Professor of Mathematics at Harvard University. In April 2022, Yau announced retirement from Harvard to become Chair Professor of mathem ...

conjectured that a particular Lorentzian version of Cheeger and Gromoll's theorem should hold. Proofs in various levels of generality were found by Jost Eschenburg, Gregory Galloway, and Richard Newman. In these results, the role of geodesic completeness is replaced by either the condition of

global hyperbolicity In mathematical physics, global hyperbolicity is a certain condition on the causal structure of a spacetime manifold (that is, a Lorentzian manifold). It's called hyperbolic because the fundamental condition that generates the Lorentzian manifold ...

or of ''timelike'' geodesic completeness. The nonnegativity of Ricci curvature is replaced by the ''timelike convergence condition'' that the Ricci curvature is nonnegative in all timelike directions. The geodesic line is required to be timelike.

References

Notes. Historical articles. * * * * * Textbooks. * * * * {{refend Theorems in Riemannian geometry