Differential geometry

A sphere (from Ancient Greek , ) is a surface analogous to the circle, a curve. In solid geometry, a sphere is the set of points that are all at the same distance from a given point in three-dimensional space. That given point is the center of the sphere, and the distance is the sphere's radius. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians.

branch of mathematics dealing with functions and geometric structures on differentiable manifolds

220px|right|thumb|Tangent to a curve. The red line is tangential to the curve at the point marked by a red dot. 220px|right|thumb|Tangent plane to a sphere

thumb|Klein quartic with 28 geodesics (marked by 7 colors and 4 patterns) In geometry, a geodesic () is a curve representing in some sense the locally shortest path (arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. It is a generalization of the notion of a "straight line".

point on a continuously differentiable plane curve at which the curve crosses its tangent, that is, the curve changes from being concave to convex, or vice versa

thumb|Two involutes (red) of a parabola

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

real smooth manifold equipped with a Riemannian metric

vector space associated to a point in a smooth manifold, consisting of vectors tangent to it (in some embedding into Euclidean space)

thumb|The blue parabola is the involute of the red curve. The red curve is the evolute of the blue parabola, and can be constructed as the locus of all centers of curvature of the blue parabola. right|thumb|The evolute of a curve (in this case, an ellipse) is the envelope of its normals.

subject of cosmology

feature of a system that is preserved under some transformation

Riemannian manifold with SU(n) holonomy

surface through every point of which runs a straight line which equally is on the surface

totally antisymmetric tensor field; section of exterior powers of the cotangent bundle

instantaneous rate of change of the function

family of curves in geometry

surface that locally minimizes its area

tensor field in general relativity and geometry

product of the principal curvatures of a surface

assignment of a tensor continuously varying across a mathematical space

radius of a circle which best approximates a curve in a given point

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)

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

number of times a curve wraps around a point in the plane

specification of derivatives along tangent vectors of a manifold

circle of immediate corresponding curvature of a curve at a point

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

maximally symmetric Lorentzian manifold with positive cosmological constant

formulas in differential geometry

differentiable function whose derivative is everywhere injective

mathematical theory in which noncommutative algebraic structures are interpreted as rings of functions on an abstract space

geometrical idea of transporting data along a curve or family of curves in a parallel and consistent manner

construct allowing differentiation of tangent vector fields of manifolds

derivative of a tensor field along the flow defined by a vector field

theoretical physics principle

in differential geometry, an extrinsic measure of curvature of a surface

curve that is a parametric solution to an initial-value problem given by a vector field

manifold with an atlas of charts to the open unit disk in ℂⁿ, such that the transition maps are holomorphic

(Riemannian geometry)

mathematical measure of how much a curve twists

number of positive, negative and zero eigenvalues of a metric tensor

point at a distance from the curve equal to the radius of curvature lying on the normal vector

fiber bundle whose fibers are group torsors (groups with the identity element forgotten)

fiber bundle of the 3-sphere over the 2-sphere, with 1-spheres as fibers

thumb|The imaginary part of the complex logarithm. Trying to define the complex logarithm on \C-\{0\} gives different answers along different paths. This leads to an infinite cyclic monodromy group and a covering of \C-\{0\} by a [[helicoid (an example of a Riemann surface).]]

mathematics term

(1,2)-tensor field associated to an affine connection; characterizes "twist" of geodesics; if nonzero, geodesics will be helices

plane in a Euclidean space or affine space which meets a submanifold at a point in such a way as to have a second order of contact at the point

An instanton (or pseudoparticle) is a notion appearing in theoretical and mathematical physics. An instanton is a classical solution to equations of motion with a finite, non-zero action, either in quantum mechanics or in quantum field theory. More precisely, it is a solution to the equations of motion of the classical field theory on a Euclidean spacetime.

In mathematics, a Grassmannian \mathbf{Gr}_k(V), also known as a Grassmann manifold, is a differentiable manifold that parameterizes the set of all k-dimensional linear subspaces of an n-dimensional vector space V over a field K that has a differentiable structure. For example, the Grassmannian \mathbf{Gr}_1(V) is the space of lines through the origin in V, so it is the same as the projective space \mathbf{P}(V) of one dimension lower than V. When V is a real or complex vector space, Grassmannians are compact smooth manifolds, of dimension k(n-k). In general they have the structure of a nonsin

quadratic form related to curvatures of surfaces

mathematical concept

study of angle-preserving transformations of a geometric space

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

study of curves from a differential point of view

complex numbers with positive imaginary part

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