Total curvature of foliations
Let \(\mathcal F\) be a one-dimensional foliation (with isolated singularities and tangent to the boundary) of a 2-dimensional Riemannian manifold \(M\) (possibly with the boundary \(\partial M)\), \(K(\mathcal F)\) -- the integral over \(M\) of the (absolute value of) geodesic curvature of the leaves of \(\mathcal F\), and \(\mathcal F^ \perp\) -- the foliation of \(M\) by curves orthogonal to \(\mathcal F\). Among some other results, the following estimates are obtained: (1) if \(M\) is a planar topological disc, then \(K(\mathcal F) \geq I(\partial M) - 2d\), where \(d\) is the diameter of \(M\) with respect to the distance \(d(x,y)\) defined as the largest lower bound for \(I(\gamma)\), \(\gamma\) being a curve in \(M\) joining \(x\) to \(y\), (2) if \(M\) is a planar topological annulus and \(\mathcal F\) has no singularities, then \(K(\mathcal F) \geq I(\partial M) - 2I(C')\), \(C'\) being the boundary or the convex hull of the bounded component of \(\mathbb{R}^ 2 \setminus M\), (3) if \(M\) is a closed hyperbolic surface, then \(K(\mathcal F) + K({\mathcal F}^ \perp) \geq (2 \pi^ 2/5I) \cdot | \chi(M) |\), \(\chi(M)\) being the Euler characteristic and \(I\) -- the length of the side of a regular pentagon in the hyperbolic plane, with interior angles \(\pi /2\).
one-dimensional foliation of a 2-dimensional Riemannian manifold
scientific article; zbMATH DE number 561227
scientific article