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, 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. It is named for John Lighton Synge, who proved it in

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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 sectional curvature. The theorem asserts: * If is even-dimensional and orientable, 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 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; 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: let be a finite-dimensional real inner product space with an orthogonal linear map with an eigenvector with eigenvalue one. If the determinant 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 later rephrased the proof so as to establish fixed points of isometries, rather than topological properties of the underlying manifold.

References

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