Riemannian geometry

intersection of the sphere and a plane which passes through the center point of the sphere

branch of differential geometry dealing with (generalized) Riemannian manifolds

thumb|upright=1.4|A Function composition|composition of two opposite isometries is a direct isometry. A reflection in a line is an opposite isometry, like (reflection w.r.t the center diagonal line) or (reflection w.r.t the right diagonal line) on the image. Translation is a direct isometry: a rigid motion.

shorthand notation for tensor operations

mathematical function which preserves angles

symmetric rank (0, 2) tensor field on a smooth manifold

real smooth manifold equipped with a Riemannian metric

in Riemannian geometry, the coefficient of the Levi-Civita connection of a Riemannian metric

tensor field in general relativity and geometry

"Remarkable theorem" about Gauss' curvature as an invariant

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)

in differential geometry, a map between two fibres of a fibre bundle induced by a path between the points in the path space connecting the two fibres and a connection in the fibre bundle

manner in which a geometric object varies with a change of basis

specification of derivatives along tangent vectors of a manifold

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

scalar quantity constructed out of second derivatives of a (pseudo-)Riemannian metric

the unique torsion-free affine connection that preserves a given (pseudo-)Rimannian metric

vector field on a (pseudo-)Riemannian manifold that preserves the metric

(Riemannian geometry)

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

theorem that closed 3-manifolds uniquely decompose into pieces with 1 of 8 types of geometric structure

quadratic form related to curvatures of surfaces

differential operator

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

spatial geometry which is not "flat" or Euclidean

in differential geometry, a function that maps each point in a surface to its normal direction

theorem

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

linear map from p-forms on an n-dimensional manifold to (n−p)-forms

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

measure of the curvature of a pseudo-Riemannian manifold

smooth manifold equipped with a Minkowski functional at each tangent space

line segment of infinitesimally small length

operator in differential topology

problem in differential geometry

Vector field in Riemannian geometry

geometric structure

unique existence of the Levi-Civita connection

seven-dimensional Riemannian manifold with holonomy group contained in G₂

field in mathematics

metric tensor describing constant negative (hyperbolic) curvature

pseudo-Riemannian manifold whose symmetry group contains an inversion symmetry about every point

totally symmetric tensor field such that the total symmetrization of its covariant derivative vanishes

a notion for convergence of metric spaces

special coordinate system in differential geometry

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

genus-7 Hurwitz surface

Bending of trajectories in general relativity by a tidal force

Fundamental formulas linking the metric and curvature tensor of a manifold

smooth map that is a critical point of the Dirichlet energy functional

type of generalization of a Riemannian manifold

theorem stating that a complete, simply-connected, n-dimensional Riemannian manifold with sectional curvature taking values in the interval (1, 4] is homeomorphic to the n-sphere

rank-3 tensor defined for a 3d (pseudo-)Riemannian manifold which measures the degree to which it fails to be conformally flat

vector bundle whose fibers carry the structure of a Clifford algebra

second-order tensor