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Given a metric on a Lorentzian manifold, the Weyl tensor for this metric may be computed.
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Given and metric
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 metric spaces ( X, d < sub > 1 </ sub >) and ( Y, d < sub > 2 </ sub >), a function f: X → Y is called uniformly continuous if for every real number ε > 0 there exists δ > 0 such that for every x, y ∈ X with d < sub > 1 </ sub >( x, y ) < δ, we have that d < sub > 2 </ sub >( f ( x ), f ( y )) < ε.
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 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 and on
* 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 the complexity of the calculations involved and the convoluted structure that a convertible bond can have, an arbitrageur often relies on sophisticated quantitative models in order to identify bonds that are trading cheap versus their theoretical value.
Given this, the bidding is said to start at the one-level when contracting for a total of seven tricks, at the two-level for eight tricks and so on to the seven-level to contract to take all thirteen tricks.
Given the federal nature of the holiday, celebrating Canada Day can be a cause of friction in the province of Quebec, where the holiday is overshadowed by Quebec's National Holiday, on June 24.
Given the rugged terrain of Ngazidja and Nzwani, and the dedication of extensive tracts to agriculture on all three islands, population pressures on Comoros are becoming increasingly critical.
Given an asymmetric subunit on a triangular face of a regular isosahedron, with three subunits per face 60 such subunits can be placed in an equivalent manner.
Given this kind of data, the estimated coefficient on Years of Education in the equation above reflects both the effect of education on wages and the effect of other variables on wages, if those other variables were correlated with education.
* Frederick Douglass lecture on Haiti – Given at the World's Fair in Chicago, January 1893.
Given also a measure on set, then, sometimes also denoted or, has as its vectors equivalence classes of measurable functions whose absolute value's-th power has finite integral, that is, functions for which one has
Given the difficulty of finding exact solutions, Einstein's field equations are also solved frequently by numerical integration on a computer, or by considering small perturbations of exact solutions.
Given an equilateral triangle, the counterclockwise rotation by 120 ° around the center of the triangle " acts " on the set of vertices of the triangle by mapping every vertex to another one.
Given these, a new term Anthropocene, is specifically proposed and used informally for the latest part of this epoch since approximately synchronous lithospheric evidence, or more recently atmospheric evidence, of human impacts have been found on the Earth and its ecosystems ; these impacts may be considered of global significance for future evolution of living species.
Given that this etymology does not appear in the usual scholarly works on etymology, this claim is usually dismissed as a false etymology.
Given the dual status of science as objective knowledge and as a human construct, good historiography of science draws on the historical methods of both intellectual history and social history.
Given a binary operation ★ on a set S, an element x is said to be idempotent ( with respect to ★) if
Given its heavy dependence on imported energy, Japan has aimed to diversify its sources.
Along with a hand-picked escort led by Given Campbell, Davis and his wife were captured on May 10, 1865, at Irwinville in Irwin County, Georgia.
Given the repeated failures and frustrations of workers ' revolutions and movements, Marx also sought to understand capitalism, and spent a great deal of time in the reading room of the British Museum studying and reflecting on the works of political economists and on economic data.
Given that estimation is undertaken on the basis of a least squares analysis, estimates of the unknown parameters β < sub > j </ sub > are determined by minimising a sum of squares function
Given that spinning a Dyson Sphere would result in the atmosphere pooling around the equator, the Ringworld removes all the extraneous parts of the structure, leaving a spinning band landscaped on the sun-facing side, with the atmosphere and inhabitants kept in place through centrifugal force and 1000 mile high perimeter walls ( rim walls ).
Given and Lorentzian
Given a Lorentzian manifold, we can find infinitely many frame fields, even if we require additional properties such as inertial motion.
Given and manifold
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 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 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 an oriented manifold M of dimension n with fundamental class, and a G-bundle with characteristic classes, one can pair a product of characteristic classes of total degree n with the fundamental class.
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 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 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.
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