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The basic regularity theorem concerning the metric in harmonic coordinates is that if the components of the metric are in the Hölder space C k, α when expressed in some coordinate chart, regardless of the smoothness of the chart itself, then the transition function from that coordinate chart to any harmonic coordinate chart will be in the ...
The harmonic coordinate condition is one of several coordinate conditions in general relativity, which make it possible to solve the Einstein field equations. A coordinate system is said to satisfy the harmonic coordinate condition if each of the coordinate functions x α (regarded as scalar fields) satisfies d'Alembert's equation .
The metric tensor is a central object in general relativity that describes the local geometry of spacetime (as a result of solving the Einstein field equations). Using the weak-field approximation, the metric tensor can also be thought of as representing the 'gravitational potential'. The metric tensor is often just called 'the metric'.
Therefore, the geometry of the 5th dimension studies the invariant properties of such space-time, as we move within it, expressed in formal equations. [11] Fifth dimensional geometry is generally represented using 5 coordinate values (x,y,z,w,v), where moving along the v axis involves moving between different hyper-volumes .
Since giving the Einstein tensor does not fully determine the Riemann tensor, but leaves the Weyl tensor unspecified (see the Ricci decomposition), the Einstein equation may be considered a kind of compatibility condition: the spacetime geometry must be consistent with the amount and motion of any matter or non-gravitational fields, in the ...
In differential geometry, one is interested in constructing minimal submanifolds of a given ambient space (,).Harmonic morphisms are useful tools for this purpose. This is due to the fact that every regular fibre ({}) of such a map : (,) (,) with values in a surface is a minimal submanifold of the domain with codimension 2. [1]
The Einstein tensor is built up from the metric tensor and its partial derivatives; thus, given the stress–energy tensor, the Einstein field equations are a system of ten partial differential equations in which the metric tensor can be solved for.
Eells and Sampson introduced the harmonic map heat flow and proved the following fundamental properties: Regularity. Any harmonic map heat flow is smooth as a map (a, b) × M → N given by (t, p) ↦ f t (p). Now suppose that M is a closed manifold and (N, h) is geodesically complete. Existence.