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      Commutative projective flows

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          Abstract

          Let \(\mathbf{x}=(x,y)\). For \(\phi(\mathbf{x})=(u(x,y),v(x,y))\) and \(z\in\mathbb{C}\), we put \(\phi^{z}=z^{-1}\phi(\mathbf{x}z)\). A two-dimensional projective flow is a solution to the projective translation equation \(\phi^{z+w}=\phi^{z}\circ\phi^{w}\), \(z,w\in\mathbb{C}\). This very specific dependence of the flow on the time parameter \(z\) is what makes projective flows a separate area of research. Let \(\phi\) and \(\psi\) be two smooth projective flows, and let \((\varpi,\varrho)\) be a vector field of \(\phi\). In this paper we explore when \(\phi\circ\psi\) is a projective flow again. We prove that this happens exactly when \(\phi\) and \(\psi\) commute, and this property is intrinsic to projective flows. From differential geometry it is known that commuting of flows is equivalent to the vanishing of Lie bracket of corresponding vector fields. All flows with \(y\varpi-x\varrho\equiv 0\) commute. However, if this is not the case, in a projective flow setting vanishing of Lie bracket is tantamount to a homogeneous system of two linear first order ODEs, so it has exactly two linearly independent solutions. And reversely: for any given smooth projective flow \(\phi\), \(y\varpi-x\varrho\neq 0\), there exists a projective flow \(\psi\) such that all flows which commute with \(\phi\) are given by \(\phi^{z}\circ\psi^{w}\), \(z,w\in\mathbb{C}\). As an application of our previous results and as the main result of this paper, we classify all commuting pairs \((\phi,\psi)\) of projective flows such that both of them have rational vector fields. It appears that this necessarily implies that both of them are algebraic flows.

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          Author and article information

          Journal
          2015-07-27
          2015-09-07
          Article
          1507.07457
          ffb9895c-d4b5-4c24-b9f9-671991268145

          http://arxiv.org/licenses/nonexclusive-distrib/1.0/

          History
          Custom metadata
          Primary 39B12, 14H05, 53AXX, 57R25. Secondary 35F05
          15 pages
          math.AG math.CA math.DG

          Geometry & Topology
          Geometry & Topology

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