Flashcards: Differential geometry

In mathematics, a differential operator is an operator defined as a function of the differentiation operator.

In the mathematical disciplines of topology, geometry, and geometric group theory, an orbifold (for "orbit-manifold") is a generalization of a manifold.

A parallel of a curve is the envelope of a family of congruent circles centered on the curve.

In the branch of mathematics called differential geometry, an affine connection is a geometric object on a smooth manifold which connects nearby tangent spaces, and so permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space.

In mathematics, particularly topology, one describes a manifold using an atlas.

In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic.

In mathematics, the Maurer–Cartan form for a Lie group G is a distinguished differential one-form on G that carries the basic infinitesimal information about the structure of G.

A sphere (from Greek σφαῖρα — sphaira, "globe, ball") is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball, (viz., analogous to a circular object in two dimensions).

In physics, a symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is preserved or remains unchanged under some transformation.

In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point.

In mathematics, physics, and engineering, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold).

Differential geometry of curves is the branch of geometry that deals with smooth curves in the plane and in the Euclidean space by methods of differential and integral calculus.

In mathematics, the directional derivative of a multivariate differentiable function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity specified by v.

Arthur Besse is a pseudonym chosen by a group of French differential geometers, led by Marcel Berger, following the model of Nicolas Bourbaki.

The shape of the universe is the and of the Universe, in terms of both curvature and topology (though, strictly speaking, the concept goes beyond both).

In mathematics, particularly in differential topology, there are two Whitney embedding theorems, named after Hassler Whitney: * The strong Whitney embedding theorem states that any smooth real m-dimensional manifold (required also to be Hausdorff and second-countable) can be smoothly embedded in the real 2m-space (R2m), if m > 0.

A multivector is the result of a product defined for elements in a vector space V.

In physics, a covariant transformation is a rule that specifies how certain entities, such as vectors or tensors, change under a change of basis.

In mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms.

In mathematics, the Teichmüller space T(S) of a (real) topological (or differential) surface S, is a space that parameterizes complex structures on S up to the action of homeomorphisms that are isotopic to the identity homeomorphism.

In mathematics, conformal geometry is the study of the set of angle-preserving (conformal) transformations on a space.

In differential geometry, a projective connection is a type of Cartan connection on a differentiable manifold.

In mathematics, the Riemannian connection on a surface or Riemannian 2-manifold refers to several intrinsic geometric structures discovered by Tullio Levi-Civita, Élie Cartan and Hermann Weyl in the early part of the twentieth century: parallel transport, covariant derivative and connection form .

In mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of the Cartesian product X × G of a space X with a group G.

In mathematics, a Hermitian symmetric space is a Hermitian manifold which at every point has as an inversion symmetry preserving the Hermitian structure.

In differential geometry, Cayley's ruled cubic surface is the surface

In differential geometry, the notion of torsion is a manner of characterizing a twist or screw of a moving frame around a curve.

In mathematics, a generalized flag variety (or simply flag variety) is a homogeneous space whose points are flags in a finite-dimensional vector space V over a field F.

In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields.

In mathematics, an isotropic manifold is a manifold in which the geometry does not depend on directions.

In mathematics, in particular in differential geometry, mathematical physics, and representation theory a Weitzenböck identity, named after Roland Weitzenböck, expresses a relationship between two second-order elliptic operators on a manifold with the same leading symbol.

Clairaut's relation, named after Alexis Claude de Clairaut, is a formula in classical differential geometry.