On the Length of Loops Generating Holonomy Groups

Holonomic Spaces

“…Although the function a → L(a) is in general not even upper-semicontinuous when regarded as a function on the holonomy group with the subspace topology (or even its Lie group topology), as pointed out by Wilkins [20], the following results gives a more positive outcome. Theorem 3.26.…”

“…that of a Lie group) of the holonomy group, this group-norm is not even upper semicontinuous. Wilkins [20] had already noted this (an immediate example is to consider a metric that is flat in a neighborhood of a point and consider the group-norm associated at that point). He proved that if the Lie group topology is compact then the group-norm topology is bounded, which is a surprising result given that the group-norm topology is finer.…”

“…The group-norm in Theorem C is given by (0.1). The study of this group-norm was already hidden in the work of Tapp [18] and Wilkins [20].…”

Recent developments in δ-Casorati curvature invariants

“…In [111], Tripathi also established the corresponding inequalities for algebraic δ -Casorati curvatures.…”

Connected holonomy is lower semicontinuous

“…Unfortunately, this is wrong i.g., which can be seen by examples of manifolds that are flat in a neighborhood of p. (Conversely, we can prove that λ is lower semi-continuous by an easy application of the fact that for each f : A → R continuous and g : A → C continuous and f bounded on preimages of g, we have µ : c → inf{f (a)|g(a) = c} is lower semi-continuous). What helps us in the end is Wilkins' result [7] that ∀g ∈ Met(M )∃L ∈ R∀A ∈ Hol(g)∃c ∈ Ω(M) : l(c) ≤ L ∧ P g (c) = A.…”