Título:  On the space of C¹ regular curves on sphere with constrained curvature

Palestrante: Cong Zhou (UFF)

Abstract: We prove that the C⁰ and C¹ topologies are the same on the set of C¹
regular curves in the 2-sphere whose tangent vectors are Lipschitz continuous, and the
a.e. existing geodesic curvatures are essentially bounded in an open interval.

    Besides, we study the subset consisting of curves that start and end at given points
with given directions, and prove that this subset is a Banach manifold.

    Furthermore, we study C¹ regular curves in the 2-sphere that start and end at given
points with given directions, whose tangent vectors are Lipschitz continuous, and their
a.e. existing geodesic curvatures have essentially bounds in an open interval. Especially,
we show that a C¹ regular curve is such a curve if and only if the infimum of its lower
curvature and the supremum of its upper curvature are constrained in the same interval.

References:
[1] C. Zhou On the space of C¹ regular curves on sphere with constrained curvature. Results in Mathematics, v. 76, p. 223, 2021.
[2] C. Zhou The geometry of C¹ regular curves in sphere with constrained curvature. J. Geom. Anal. 31 (2020), no. 6, 5974–5987.