Differential geometry

in geometry, transferring a differential form or fiber bundle from the codomain of a continuous map to the domain

spatial geometry which is not "flat" or Euclidean

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

angle between the tangent line to the curve at the given point and the x-axis

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

Mathematical structure in differential geometry

mathematical concept of a compact manifold without boundary

alt=Visualisation of parallel transport on a sphere|thumb|Parallel transport on a sphere along a piecewise smooth path. The initial vector is labelled as V, parallel transported along the curve, and the resulting vector is labelled as \mathcal{P}_{\gamma}(V). The outcome of parallel transport will be different if the path is varied.

generalization of the concept of parallel lines

linear approximation of smooth maps on tangent spaces

on a Lie group G, a canonical 1-form valued in its own Lie algebra; the unique principal-bundle connection on the unique G-bundle over the one-point space

In physics and mathematics, a pseudotensor is usually a quantity that transforms like a tensor under an orientation-preserving coordinate transformation (e.g. a proper rotation) but additionally changes sign under an orientation-reversing coordinate transformation (e.g., an improper rotation), which is a transformation that can be expressed as a proper rotation followed by reflection. This is a generalization of a pseudovector. To evaluate a tensor or pseudotensor sign, it has to be contracted with some vectors, as many as its rank is, belonging to the space where the rotation is made while ke

A generalization of a basis to sets of possibly linearly dependent vectors which also satisfy the frame condition

operator in differential topology

Lie-algebra-valued two-form that describes the curvature of a principal bundle

a type of smooth complex surface of Kodaira dimension 0

smooth manifold equipped with a Minkowski functional at each tangent space

parametrizes complex structures on a surface

vector bundle, complementary to the tangent bundle, associated to an embedding

nonlinear hyperbolic partial differential equation in 1 + 1 dimensions

operation that takes a differentiable function f and produces a polynomial, the truncated Taylor polynomial of f, at each point of its domain

system with kinematic constraints that cannot be integrated in position-level constraints

seven-dimensional Riemannian manifold with holonomy group contained in G₂

secondary characteristic class defined for odd-dimensional manifolds with G-bundles with connection; in 2n−1 dimensions, defined as (formal) exterior antiderivative of tr(Fⁿ) where F is the curvature of the connection

binary operation between a vector field and a differential form

math/physics concept

mathematical discipline at the interface of differential geometry and differential equations

envelope of rays either reflected or refracted by a manifold

equivalence between the derived Fukaya category of a symplectic manifold and the derived category of coherent sheaves on a complex manifold

isometry of a metric space

Special product of manifolds

physics concept

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

field in mathematics

Mathematical vector components

right|frame|The Trott curve (black) has 28 real bitangents (red). This image shows 7 of them; the others are symmetric with respect to 90° rotations through the origin and reflections through the two blue axes.

Natural moving frame in differential geometry of surfaces

Periodic minimal surface

open 3-manifold that is contractible, but not homeomorphic to R³

concepts from applied mathematics, computer science and engineering to design efficient algorithms for complex 3D models.

generalization of affine connections

Ehresmann connection on a principal bundle that is compatible with the group action

formula in differential geometry

shortest paths on a bounded deformed sphere-like quadric surface

classification of the possible algebraic symmetries of the Weyl tensor at each point in a Lorentzian manifold

subbundle of the tangent bundle

Result in integral geometry

formula in classical differential geometry

On when a set of compact Riemannian manifolds of a given dimension is relatively compact

Construction for minimal surfaces

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

group of gauge symmetries in a gauge theory

the manifold of all orthonormal k-frames in n-dimensional Euclidean space

tensor index notation for tensor-based calculations

Surface homeomorphic to a sphere