closed manifold
compact manifold without boundary
n-sphere
generalization of the ordinary sphere to spaces of arbitrary dimension
essential manifold
compact manifold M whose fundamental class [M] defines a nonzero element under the natural homomorphism Hₙ(M) → Hₙ(K(π₁(M),1))
torus
doughnut-shaped surface of revolution, based on a revolved figure of a circle
almost flat manifold
a smooth compact manifold such that, for any ε>0, there is a Riemannian metric g such that the diameter (measured using g) is at most one and such that the absolute value of sectional curvature is at most ε everywhere
genus g surface
smooth closed surface with g holes
complex projective space
the space of lines in a complex vector space
real projective space
type of topological space