Flow (geometry)

In mathematics, a flow formalizes the idea of the motion of particles in a fluid. Flows are ubiquitous in science, including engineering and physics. The notion of flow is basic to the study of ordinary differential equations. Informally, a flow may be viewed as a continuous motion of points over time. More formally, a flow is a group action of the real numbers on a set.

The idea of a vector flow, that is, the flow determined by a vector field, occurs in the areas of differential topology, Riemannian geometry and Lie groups. Specific examples of vector flows include the geodesic flow, the Hamiltonian flow, the Ricci flow, the mean curvature flow, and the Anosov flow. Flows may also be defined for systems of random variables and , and occur in the study of ergodic dynamical systems. The most celebrated of these is perhaps the Bernoulli flow.

A flow on a set X is a group action of the additive group of real numbers on X. More explicitly, a flow is a mapping

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