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 ...

, specifically

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 ...

, Synge's theorem is a classical result relating the

curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the canonic ...

of a

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 ...

to its

topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

. It is named for

John Lighton Synge John Lighton Synge (; 23 March 1897 – 30 March 1995) was an Irish mathematician and physicist, whose seven-decade career included significant periods in Ireland, Canada, and the USA. He was a prolific author and influential mentor, and is cre ...

, who proved it in

1936 Events January–February * January 20 – George V of the United Kingdom and the British Dominions and Emperor of India, dies at his Sandringham Estate. The Prince of Wales succeeds to the throne of the United Kingdom as King E ...

Theorem and sketch of proof

Let be a

closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...

Riemannian manifold with

positive Positive is a property of positivity and may refer to: Mathematics and science * Positive formula, a logical formula not containing negation * Positive number, a number that is greater than 0 * Plus sign, the sign "+" used to indicate a posit ...

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 theorem asserts: * If is even-dimensional and

orientable In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...

, then is

simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...

. * If is odd-dimensional, then it is orientable. In particular, a closed manifold of even dimension can support a positively curved Riemannian metric only if its

fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...

has one or two elements. The proof of Synge's theorem can be summarized as follows. Given a geodesic with an orthogonal and parallel vector field along the geodesic (i.e. a parallel section of the

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 ...

to the geodesic), then Synge's earlier computation of the ''second variation formula'' for arclength shows immediately that the geodesic may be deformed so as to shorten its length. The only tool used at this stage is the assumption on sectional curvature. The construction of a parallel vector field along any path is automatic via

parallel transport In geometry, parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection (vector bundle), c ...

; the nontriviality in the case of a loop is whether the values at the endpoints coincide. This reduces to a problem of pure

linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. ...

: let be a finite-dimensional real

inner product space In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often den ...

with an orthogonal linear map with an eigenvector with eigenvalue one. If the

determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...

of is positive and the dimension of is even, or alternatively if the determinant of is negative and the dimension of is odd, then there is an eigenvector of with eigenvalue one which is orthogonal to . In context, is the tangent space to at a point of a geodesic loop, is the parallel transport map defined by the loop, and is the tangent vector to the geodesic. Given any noncontractible loop in a complete Riemannian manifold, there is a representative of its (free) homotopy class which has minimal possible arclength, and it is a geodesic. According to Synge's computation, this implies that there cannot be a parallel and orthogonal vector field along this geodesic. However: * Orientability implies that the parallel transport map along every loop has positive determinant. Even-dimensionality then implies the existence of a parallel vector field, orthogonal to the geodesic. * Non-orientability implies the non-contractible loop can be chosen so that the parallel transport map has negative determinant. Odd-dimensionality then implies the existence of a parallel vector field, orthogonal to the geodesic. This contradiction establishes the non-existence of noncontractible loops in the first case, and the impossibility of non-orientability in the latter case.

Alan Weinstein Alan David Weinstein (17 June, 1943, New York City) is a professor of mathematics at the University of California, Berkeley, working in the field of differential geometry, and especially in Poisson geometry. Education and career Weinstein ob ...

later rephrased the proof so as to establish fixed points of

isometries In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...

, rather than topological properties of the underlying manifold.

References

Sources. * * * * Theorems in Riemannian geometry {{differential-geometry-stub