Regularity of a radial field on a Hadamard manifold

Research Activities on Magnesium Alloys at Seoul National University, Korea

“…Hyperbolic space clearly satisfies these regularity conditions, and in Proposition 5.1, we show that the radial fields are smooth on this space. Shcherbakov (1983) showed that if the norm of the curvature tensor R is bounded and, again, ∇R is bounded and ∆ < 0, the radial fields are C 2 . This is a stronger result than that of Green (1974) because the boundedness of the sectional curvatures implies the boundedness of R (the second theorem in Karcher (1970) gives an explicit bound for the norm of R in terms of δ and ∆).…”

“…In fact, though they are known to be C 1 (see Proposition 3.1 of Heintze and Im Hof (1977)) they are not even guaranteed to be C 2 . Green (1974) and Shcherbakov (1983) provide some conditions on the curvature of the manifold under which twice continuous differentiability can be guaranteed, but since they require the supremum of the sectional curvatures to be less than 0, these results do not apply to P m . However, we can show that the radial fields are, in fact, smooth in P m , just as Shin and Oh (2023) did in hyperbolic spaces.…”

“…x (0) in T x M with ξ. The vector fields on M defined by x → ξ x for ξ ∈ ∂M , which have been called radial fields (Heintze and Im Hof (1977), Shcherbakov (1983)), are also the negative gradients of the so-called Busemann functions x → lim t→∞ d(x, γ(t)) − t,…”