Given a sphere of unit radius, place its center at the origin of the complex plane, oriented so that the equator on the sphere coincides with the unit circle in the plane, and the north pole is " above " the plane.

Given two oriented submanifolds of complementary dimensions in a simply connected manifold of dimension, one can apply an isotopy to one of the submanifolds so that all the points of intersection have the same sign.

Given a fixed oriented line L in the Euclidean plane R < sup > 2 </ sup >, a meander of order n is a non-self-intersecting closed curve in R < sup > 2 </ sup > which transversally intersects the line at 2n points for some positive integer n. Two meanders are said to be equivalent if they are homeomorphic in the plane.

# Given any two reduced alternating diagrams D < sub > 1 </ sub > and D < sub > 2 </ sub > of an oriented, prime alternating link: D < sub > 1 </ sub > may be transformed to D < sub > 2 </ sub > by means of a sequence of certain simple moves called flypes.

Given a volume form ω on an oriented manifold, the density | ω | is a volume pseudo-form on the nonoriented manifold obtained by forgetting the orientation.

Given any two reduced alternating diagrams D < sub > 1 </ sub > and D < sub > 2 </ sub > of an oriented, prime alternating link: D < sub > 1 </ sub > may be transformed to D < sub > 2 </ sub > by means of a sequence of certain simple moves called flypes.

Given a subset X of a manifold M and a subset Y of a manifold N, a function f: X → Y is said to be smooth if for all p in X there is a neighborhood of p and a smooth function g: U → N such that the restrictions agree ( note that g is an extension of f ).

Given both Einstein's equations and suitable equations for the properties of matter, such a solution consists of a specific semi-Riemannian manifold ( usually defined by giving the metric in specific coordinates ), and specific matter fields defined on that manifold.

Given a differentiable manifold, one can unambiguously define the notion of tangent vectors and then directional derivatives.

Given a differentiable manifold M, a vector field on M is an assignment of a tangent vector to each point in M. More precisely, a vector field F is a mapping from M into the tangent bundle TM so that is the identity mapping

Given any coordinate chart about some point on the manifold, the above identities may be written in terms of the components of the Riemann tensor at this point as:

Given a local coordinate system x < sup > i </ sup > on the manifold, the reference axes for the coordinate system are the vector fields

Given an orientable Haken manifold M, by definition it contains an orientable, incompressible surface S. Take the regular neighborhood of S and delete its interior from M. In effect, we've cut M along the surface S. ( This is analogous, in one less dimension, to cutting a surface along a circle or arc.

Given a Riemannian manifold and two linearly independent tangent vectors at the same point, u and v, we can define

Given a Riemannian manifold with metric tensor, we can compute the Ricci tensor, which collects averages of sectional curvatures into a kind of " trace " of the Riemann curvature tensor.

Given a complex hermitian vector bundle V of complex rank n over a smooth manifold M,

Given a manifold and a Lie algebra valued 1-form, over it, we can define a family of p-forms:

Given a manifold M representing ( continuous / smooth / with certain boundary conditions / etc.

* Given the action of a Lie algebra g on a manifold M, the set of g-invariant vector fields on M is a Lie algebroid over the space of orbits of the action.

* Given any manifold, there is a Lie groupoid called the pair groupoid, with as the manifold of objects, and precisely one morphism from any object to any other.

* Given a Lie group acting on a manifold, there is a Lie groupoid called the translation groupoid with one morphism for each triple with.

Given a manifold with a submanifold, one sometimes says can be knotted in if there exists an embedding of in which is not isotopic to.

A more general class are flat G-bundles with for a manifold F. Given a representation, the flat-bundle with monodromy is given by, where acts on the universal cover by deck transformations and on F by means of the representation.

Given a smooth 4n-dimensional manifold M and a collection of natural numbers

Given a function on, one may " geometrize " it by taking it to define a new manifold.

Given a statistical manifold, with coordinates given by, one writes for the probability distribution.

* Given an R-module M, the endomorphism ring of M, denoted End < sub > R </ sub >( M ) is an R-algebra by defining ( r · φ )( x ) = r · φ ( x ).

Given two manifolds M and N, a bijective map f from M to N is called a diffeomorphism if both

Given an evaluation e of variables by elements of M < sub > w </ sub >, we

Given a system of n-dimensional variables ( physical variables ), in k ( physical ) dimensions, write the dimensional matrix M, whose rows are the dimensions and whose columns are the variables: the ( i, j ) th entry is the power of the ith unit in the jth variable.

Given two complexes M < sub >*</ sub > and N < sub >*</ sub >, a chain map between the two is a series of homomorphisms from M < sub > i </ sub > to N < sub > i </ sub > such that the entire diagram involving the boundary maps of M and N commutes.

Given an SVD of M, as described above, the following two relations hold:

Given a set M of molecules, chemical reactions can be roughly defined as pairs r =( A, B ) of subsets from M.

Given a Hermitian form Ψ on a complex vector space V, the unitary group U ( Ψ ) is the group of transforms that preserve the form: the transform M such that Ψ ( Mv, Mw ) = Ψ ( v, w ) for all v, w ∈ V. In terms of matrices, representing the form by a matrix denoted, this says that.

Given the morphological distinctness of the Cape Verde birds and the fact that the Cape Verde population was isolated from other populations of Red Kites, it cannot be conclusively resolved at this time whether the Cape Verde population was not a distinct subspecies ( as M. migrans fasciicauda ) or even species that frequently absorbed stragglers from the migrating European populations into its gene pool.

Given such a G-module M, it is natural to consider the subgroup of G-invariant elements:

Given a smooth curve γ on ( M, g ) and a vector field V along γ its derivative is defined by

If V is finite-dimensional, then V * has the same dimension as V. Given a basis

Given a partial isometry V, the deficiency indices of V are defined as the dimension of the orthogonal complements of the domain and range:

; Degree of an extension: Given an extension E / F, the field E can be considered as a vector space over the field F, and the dimension of this vector space is the degree of the extension, denoted by: F.

Given the affine group of an affine space A, the stabilizer of a point p is isomorphic to the general linear group of the same dimension ( so the stabilizer of a point in Aff ( 2, R ) is isomorphic to GL ( 2, R )); formally, it is the general linear group of the vector space: recall that if one fixes a point, an affine space becomes a vector space.

* Given an eigenvalue λ < sub > i </ sub >, its geometric multiplicity is the dimension of Ker ( A − λ < sub > i </ sub > I ), and it is the number of Jordan blocks corresponding to λ < sub > i </ sub >.

Given a field extension K / F, the field K can be considered as a vector space over the field F. The dimension of this vector space is the degree of the extension and is denoted by: F.

Given two maps f and g from an orientable manifold X to an orientable manifold Y of the same dimension, the Lefschetz coincidence number of f and g is defined as

Given this scaling dimension for, there are certain nonlinear modifications of massless scalar field theory which are also scale-invariant.

Given that the ZWZ focused on military aspects of the struggle, its civilian dimension was less clearly defined and developed more slowly — a situation exacerbated by the complex political discussions that were then unfolding between politicians in occupied Poland and the government in exile ( first located in Paris, and after the fall of France, in London ).

Given the small dimension of the tunnels, steam power, as used on London's other underground railways, was not feasible for a deep tube railway.

Given a connected and orientable manifold M of dimension 4k, the cup product gives rise to a quadratic form Q on the ' middle ' real cohomology group

Given below is the five point method for the first derivative ( five-point stencil in one dimension ).