Riemannian manifolds

real smooth manifold equipped with a Riemannian metric

tensor field in general relativity and geometry

2-tensor obtained as a contraction of the Riemann curvature 4-tensor on a Riemannian manifold (or, more generally, a smooth manifold equipped with affine connection)

smooth manifold equipped with nowhere degenerate (but not necessarily positive-definite) metric tensor

flow associated to the partial differential equation ∂𝑔/∂𝑡=−2Ric[𝑔] on a Riemannian manifold

smooth manifold carrying compatible complex, Riemannian, and symplectic structures

scalar quantity; given two unit tangent vectors u, v at the same point, defined as K(u,v) = ⟨R(u,v)v,u⟩/(1-⟨u,v⟩²), where R is the Riemann curvature

theorem

isomorphism between the tangent and cotangent bundles on a smooth manifold; induced by either a RIemannian or symplectic structure

top-dimensional differential form that can be defined on orientable manifolds

smooth manifold equipped with a Minkowski functional at each tangent space

unique existence of the Levi-Civita connection

Riemannian or pseudo-Riemannian differentiable manifold whose Ricci tensor is proportional to the metric

manifold modeled on Hilbert spaces; separable Hausdorff space in which each point has a neighborhood homeomorphic to an infinite dimensional Hilbert space

space where every point locally resembles a hyperbolic space

Riemannian manifold with vanishing Riemann curvature

lift of the SO(n) frame bundle of an oriented Riemannian manifold into Spin(n), the spin group

type of generalization of a Riemannian manifold

riemannian manifolds whose ricchi curvature vanishes

Riemannian manifold with Sp(n) holonomy, or equivalently with an S² family of Kähler structures