Projective flatness of cylindrically symmetric Finsler metrics

arXiv:2611.08747v1 Announce Type: new Abstract: In this paper, we study cylindrically symmetric projectively flat Finsler metrics \(F=|\bar y|\phi(x^0,r,s,z)\). We reformulate the conditions for projective flatness and strong convexity in terms of \(\Omega=\phi-s\phi_s-z\phi_z\), isolating the compatibility condition missing in earlier work. Using the method of characteristics, we give an integral formula for all smooth local solutions near points with \(r>0\) and \(s=z=0\). These solutions are determined by a characteristic potential up to the addition of an invariant exact one-form. In the reversible case, the equation for the potential reduces to a \((2+1)\)-dimensional wave equation on a half-space. We illustrate the reconstruction with a polynomial potential and compute the potentials of a known family of spherically symmetric metrics and their linear pullbacks. The convexity conditions also admit decreasing profiles and recover the spherical Beltrami metric as a particular case. Finally, suitable positive sums of linear pullbacks yield smooth reversible projectively flat Finsler metrics on \(\mathbb R^{n+1}\) whose tangent norm at the origin is non-Riemannian. Consequently, these metrics are not linear pullbacks of spherically symmetric metrics

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