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 two metric spaces ( X, d < sub > X </ sub >) and ( Y, d < sub > Y </ sub >), where d < sub > X </ sub > denotes the metric on the set X and d < sub > Y </ sub > is the metric on set Y ( for example, Y might be the set of real numbers R with the metric d < sub > Y </ sub >( x, y )

* Given a positive real number ε, an ε-isometry or almost isometry ( also called a Hausdorff approximation ) is a map between metric spaces such that

Given a coordinate system and a metric tensor, scalar curvature can be expressed as follows

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 specified distribution of matter and energy in the form of a stress – energy tensor, the EFE are understood to be equations for the metric tensor, as both the Ricci tensor and scalar curvature depend on the metric in a complicated nonlinear manner.

Given a sequence ( X < sub > n </ sub >, p < sub > n </ sub >) of locally compact complete length metric spaces with distinguished points, it converges to ( Y, p ) if for any R > 0 the closed R-balls around p < sub > n </ sub > in X < sub > n </ sub > converge to the closed R-ball around p in Y in the usual Gromov – Hausdorff sense.

Using tensor calculus, proper time is more rigorously defined in general relativity as follows: Given a spacetime which is a pseudo-Riemannian manifold mapped with a coordinate system and equipped with a corresponding metric tensor, the proper time experienced in moving between two events along a timelike path P is given by the line integral

is also true in the case of compact manifolds, due to Yau's proof of the Calabi conjecture: Given a compact, Kähler, holomorphically symplectic manifold ( M, I ), it is always equipped with a compatible hyperkähler metric.

Given the frame field, one can also define a metric by conceiving of the frame field as an orthonormal vector field.

Given the inverse of the metric tensor above, the explicit form of the kinetic energy operator in terms of Euler angles follows by simple substitution.

Given the metric η, we can ignore the covariant and contravariant distinction for T.

Given a metric space ( X, d ), or more generally, an extended pseudoquasimetric ( which will be abbreviated xpq-metric here ), one can define an induced map d: X × P ( X )→ by d ( x, A ) = inf

Given a metric space a point is called close or near to a set if

Given a metric on a Lorentzian manifold, the Weyl tensor for this metric may be computed.

Given two tangent vectors u and v at a point x in M, the metric can be evaluated on u and v to give a real number:

Given a manifold M, one looks for the longest product of systoles which give a " curvature-free " lower bound for the total volume of M ( with a constant independent of the metric ).

Given an arbitrary Riemannian metric g on an almost complex manifold M one can construct a new metric g ′ compatible with the almost complex structure J in an obvious manner:

Given two affine spaces and, over the same field, a function is an affine map if and only if for every family of weighted points in such that

Given two spaces X and Y, we say they are homotopy equivalent or of the same homotopy type if there exist continuous maps and such that is homotopic to the identity map id < sub > X </ sub > and is homotopic to id < sub > Y </ sub >.

Given two normed vector spaces V and W ( over the same base field, either the real numbers R or the complex numbers C ), a linear map A: V → W is continuous if and only if there exists a real number c such that

Given two normed vector spaces V and W, a linear isometry is a linear map f: V → W that preserves the norms:

Given a collection of spaces and maps with maps ( compatible with the inclusions, an X-structure is a lift of ν to a map.

Given a pair of spaces, for simplicity we denote.

Given a manifold Q, a vector field X on Q ( or equivalently, a section of the tangent bundle TQ ) can be thought of as a function acting on the cotangent bundle, by the duality between the tangent and cotangent spaces.

Given a pair of spaces ( X, A ) the mapping class group of the pair is the isotopy-classes of automorphisms of the pair, where an automorphism of ( X, A ) is defined as an automorphism of X that preserves A, i. e. f: X → X is invertible and f ( A )

Given two I-graded vector spaces V and W, their direct sum has underlying vector space V ⊕ W with gradation

Given a set X and an indexed family ( Y < sub > i </ sub >)< sub > i ∈ I </ sub > of topological spaces with functions

Given a continuous map there is a map defined by This makes into a functor from the category of topological spaces into itself.

Given a set and a family of topological spaces with functions

Given Borel equivalence relations E and F on Polish spaces X and Y respectively, one says that E is Borel reducible to F, in symbols E ≤< sub > B </ sub > F, if and only if there is a Borel function

Given any desired positive integer c, this theorem shows that one can find an algebraic solution approximating a formal power series solution up to the degree specified by c. This leads to theorems that deduce the existence of certain formal moduli spaces of deformations as schemes.

Given three Hilbert spaces,,

: Given any set X of pairwise disjoint non-empty sets, there exists at least one set C that contains exactly one element in common with each of the sets in X.

* Given any Banach space X, the continuous linear operators A: X → X form a unitary associative algebra ( using composition of operators as multiplication ); this is a Banach algebra.

* Given any topological space X, the continuous real-or complex-valued functions on X form a real or complex unitary associative algebra ; here the functions are added and multiplied pointwise.

Given any element x of X, there is a function f < sup > x </ sup >, or f ( x ,·), from Y to Z, given by f < sup > x </ sup >( y ) := f ( x, y ).

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 any set X, there is an equivalence relation over the set of all possible functions X → X.

Given a topological space X, let G < sub > 0 </ sub > be the set X.

Given a ( random ) sample the relation between the observations Y < sub > i </ sub > and the independent variables X < sub > ij </ sub > is formulated as

# Given any point x in X, and any sequence in X converging to x, the composition of f with this sequence converges to f ( x )

Given X such that

Given an arbitrary topological space ( X, τ ) there is a universal way of associating a completely regular space with ( X, τ ).

Given any embedding of a Tychonoff space X in a compact Hausdorff space K the closure of the image of X in K is a compactification of X.