Synge's theorem
Encyclopedia
In mathematics
, specifically Riemannian geometry
, Synge's theorem is a classical result relating the curvature
of a Riemannian manifold
to its topology
. It is named for John Lighton Synge
, who proved it in 1936. Let M be a compact
Riemannian manifold with positive sectional curvature
. The theorem asserts:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, specifically Riemannian geometry
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 which varies smoothly from point to point. This gives, in particular, local notions of angle, length...
, Synge's theorem is a classical result relating the curvature
Curvature
In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...
of a Riemannian manifold
Riemannian manifold
In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...
to its topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
. It is named for John Lighton Synge
John Lighton Synge
John Lighton Synge was an Irish mathematician and physicist.-Background:Synge was born 1897 in Dublin, Ireland, in a Protestant family and educated at St. Andrew's College, Dublin. He entered Trinity College, Dublin in 1915...
, who proved it in 1936. Let M be a compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
Riemannian manifold with positive sectional curvature
Sectional curvature
In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature K depends on a two-dimensional plane σp in the tangent space at p...
. The theorem asserts:
- If M is even-dimensional and orientable, then M is simply connected.
- If M is odd-dimensional, then it is orientable.