In differential geometry, a complete Riemannian manifold is called a Ricci soliton if, and only if, there exists a smooth vector field such that
for some constant . Here is the Ricci curvature tensor and represents the Lie derivative. If there exists a function such that we call a gradient Ricci soliton and the soliton equation becomes
Note that when or the above equations reduce to the Einstein equation. For this reason Ricci solitons are a generalization of Einstein manifolds.
complete Riemannian manifold ( M , g ) {\displaystyle (M,g)} is called a Riccisoliton if, and only if, there exists a smooth vector field V {\displaystyle...
the Ricci flow, its geometry remains the same. Such solutions are called steady Riccisolitons. An example of a 3-dimensional steady Riccisoliton is the...
analogous to the way that one obtains a multiple soliton solution of the KdV from the single soliton solution (which can be found from Lie's notion of...
integrable models, when solutions are sometimes a sort of superposition of solitons; this happens e.g. for the Korteweg–de Vries equation. It is often possible...
Commutative Algebra, Andrea Ferretti (2023, ISBN 978-1-4704-7128-6) 235 RicciSolitons in Low Dimensions, Bennett Chow (2023, ISBN 978-1-4704-7428-7) 236 Alexandrov...
This implies that the Schwarzschild black hole is a form of gravitational soliton. The spatial curvature of the Schwarzschild solution for r > rs can be...
motion with Euclidean time and finite Euclidean action. In the context of soliton theory the corresponding solution is known as a kink. In view of their...
\mathbf {v} _{x}-2\mathbf {v} \times (\mathbf {v} \times \mathbf {b} )} Solitons Boltzmann equation 1+6 ∂ f i ∂ t + p i m i ⋅ ∇ f i + F ⋅ ∂ f i ∂ p i =...
See Ricci calculus and Van der Waerden notation for the notation. In quantum field theory, the nonlinear Dirac equation is a model of self-interacting...
}-\partial _{\gamma }F_{[\alpha \beta \delta ]\varepsilon }),} and the dual Ricci curvature tensor and scalar curvature of the dual graviton become, respectively...
predict the existence of monopoles which, unlike elementary particles, are solitons (localized energy packets). The initial results of using these models to...
mechanics often becomes relevant when studying the dynamics of supersymmetric solitons, and due to the simplified nature of having fields which are only functions...
elegant proof of Lawson's conjecture, for his characterization of soliton solutions of Ricci flows and mean curvature in dimension 3 as well as for his remarkable...
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