mathematics 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 ...

, the soul theorem is a theorem of

Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to poin ...

that largely reduces the study of complete

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...

s of non-negative

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 ...

to that of the

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

case.

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 the theorem in 1972 by generalizing a 1969 result of Gromoll and Wolfgang Meyer. The related soul conjecture, formulated by Cheeger and Gromoll at that time, was proved twenty years later by

Grigori Perelman Grigori Yakovlevich Perelman ( rus, links=no, Григорий Яковлевич Перельман, p=ɡrʲɪˈɡorʲɪj ˈjakəvlʲɪvʲɪtɕ pʲɪrʲɪlʲˈman, a=Ru-Grigori Yakovlevich Perelman.oga; born 13 June 1966) is a Russian mathemati ...

Soul theorem

Cheeger and Gromoll's soul theorem states: :If is a

complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...

connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...

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 ...

with nonnegative

, then there exists a closed totally convex,

totally geodesic This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology. The following articles may also be useful; they either contain specialised vocabulary or provi ...

embedded submanifold In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which p ...

whose

normal bundle In differential geometry, a field of mathematics, a normal bundle is a particular kind of vector bundle, complementary to the tangent bundle, and coming from an embedding (or immersion). Definition Riemannian manifold Let (M,g) be a Riemannian m ...

diffeomorphic In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an Inverse function, invertible Function (mathematics), function that maps one differentiable manifold to another such that both the function and its inverse function ...

to . Such a submanifold is called a soul of . By the Gauss equation and total geodesicity, the induced Riemannian metric on the soul automatically has nonnegative sectional curvature. Gromoll and Meyer had earlier studied the case of positive sectional curvature, where they showed that a soul is given by a single point, and hence that is diffeomorphic to

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 ...

. Very simple examples, as below, show that the soul is not uniquely determined by in general. However, Vladimir Sharafutdinov constructed a 1-Lipschitz retraction from to any of its souls, thereby showing that any two souls are isometric. This mapping is known as the Sharafutdinov's retraction. Cheeger and Gromoll also posed a converse question of whether there is a complete Riemannian metric of nonnegative sectional curvature on the total space of any

vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...

over closed manifolds of positive sectional curvature. Examples. * As can be directly seen from the definition, every

manifold is its own soul. For this reason, the theorem is often stated only for non-compact manifolds. * As a very simple example, take to be

. The sectional curvature is everywhere, and any point of can serve as a soul of . * Now take the

paraboloid In geometry, a paraboloid is a quadric surface that has exactly one axis of symmetry and no center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar property of symmetry. Every plane ...

, with the metric being the ordinary Euclidean distance coming from the embedding of the paraboloid in Euclidean space . Here the sectional curvature is positive everywhere, though not constant. The origin is a soul of . Not every point of is a soul of , since there may be geodesic loops based at , in which case

\

wouldn't be totally convex. * One can also consider an infinite

cylinder A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infin ...

, again with the induced Euclidean metric. The sectional curvature is everywhere. Any "horizontal" circle with fixed is a soul of . Non-horizontal cross sections of the cylinder are not souls since they are neither totally convex nor totally geodesic.

Soul conjecture

As mentioned above, Gromoll and Meyer proved that if has positive sectional curvature then the soul is a point. Cheeger and Gromoll conjectured that this would hold even if had nonnegative sectional curvature, with positivity only required of all sectional curvatures at a single point. This soul conjecture was proved by

, who established the more powerful fact that Sharafutdinov's retraction is a

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 (' ...

, and even a submetry.

References

Sources. * * * * * * * * * {{Refend Theorems in Riemannian geometry