arXiv subjects

Jacob W. Erickson

Publications and source records attributed to Jacob W. Erickson.

7 recordsLinked to original sources

arXiv2026

A kinematic explanation of the 1:3 ratio for rolling spheres and the exceptional simple Lie group of rank two

Given a pair of (round) 2-dimensional spheres, one of which has radius three times that of the other, the Lie algebra of local infinitesimal symmetries for the distribution corresponding to rolling the spheres along each other (without letting them slip or twist) happens to be isomorphic to the split real form of the exceptional simple Lie algebra of rank 2. These exceptional local symmetries appear only for this specific 1:3 ratio of radii, however, and while there are several proofs of this result, a straightforward kinematic explanation for the seemingly miraculous appearance of an exceptional simple Lie group in this situation has long been desired. In this paper, we provide such an explanation, relating the ratio of radii to the intersections of a pair of one-parameter subgroups that can be seen from the rolling sphere perspective. The approach does not require the split-octonions, as we construct the exceptional simple Lie group directly by Tanaka prolongation to aid in visualizing the underlying geometry.

math.DG↗

arXiv2026

Inequivalence of the various notions of completeness for Cartan geometries

We demonstrate that the three prevailing notions of completeness for Cartan geometries are not actually equivalent and provide methods for constructing many examples in the affine case, modernizing and extending results due to Yeaton Clifton. In an effort to rectify the folklore on the topic, we also provide detailed explanations of how specious proofs in the literature that would claim that these types of completeness are equivalent break down.

math.DG↗

arXiv2024

Holonomy of Parabolic Geometries Near Isolated Higher-Order Fixed Points

For Cartan geometries admitting an automorphism with isotropy satisfying a particular, loosely dynamical property on their model geometries, we demonstrate the existence of an open subset of the geometry with trivial holonomy. This property, which generalizes characteristics of isotropies corresponding to isolated higher-order fixed points in parabolic geometries that are known to require a nearby open subset to have vanishing curvature, only relies upon the behavior of the isotropy in the model geometry, and therefore applies regardless of initial curvature assumptions, such as regularity or normality. Along the way to proving our main results, we also derive a couple of results for working with holonomy, relating to limits of sequences of developments and the existence of antidevelopments, that are useful in their own right. To showcase the effectiveness of the techniques developed, we use them to completely characterize all almost c-projective and almost quaternionic structures that admit a nontrivial automorphism with a higher-order fixed point, as well as all nondegenerate partially integrable almost CR structures that admit a higher-order fixed point with non-null isotropy.

math.DG↗

arXiv2023

Some elementary observations regarding reductive Cartan geometries

After defining generalizations of the notions of covariant derivatives and geodesics from Riemannian geometry for reductive Cartan geometries in general, various results for reductive Cartan geometries analogous to important elementary results from Riemannian geometry are proven using these generalizations. In particular, a generalization of the Hopf-Rinow theorem is given with a pleasantly concise proof.

math.DG↗

arXiv2023

A method for determining Cartan geometries from the local behavior of automorphisms

We introduce a construction for a Cartan geometry that captures the local behavior of a given geometric automorphism near a distinguished element. The result of this construction, which we call the sprawl generated by the automorphism, is uniquely characterized by a kind of universal property that allows us to compare different Cartan geometries that admit automorphisms with equivalent local behavior near a distinguished element. As example applications, we describe how to construct non-flat real projective structures admitting nontrivial automorphisms with higher-order fixed points and extend some known local automorphisms with higher-order fixed points on non-flat parabolic geometries to global automorphisms.

math.DG↗

arXiv2022

Higher rank parabolic geometries with essential automorphisms and nonvanishing curvature

We construct infinite families of regular normal Cartan geometries with nonvanishing curvature and essential automorphisms on closed manifolds for many higher rank parabolic model geometries. To do this, we use particular elements of the kernel of the Kostant Laplacian to construct homogeneous Cartan geometries of the desired type, giving a global realization of an elegant local construction due to Kruglikov and The, and then modify these homogeneous geometries to make their base manifolds compact. As a demonstration, we apply the construction to quaternionic contact structrures of mixed signature, among other examples.

math.DG↗

arXiv2021

Intrinsic holonomy and curved cosets of Cartan geometries

We provide an intrinsic notion of curved cosets for arbitrary Cartan geometries, simplifying the existing construction of curved orbits for a given holonomy reduction. To do this, we define an intrinsic holonomy group, which is shown to coincide precisely with the standard definition of the holonomy group for Cartan geometries in terms of an associated principal connection. These curved cosets retain many characteristics of their homogeneous counterparts, and they behave well under the action of automorphisms. We conclude the paper by using the machinery developed to generalize the de Rham decomposition theorem for Riemannian manifolds and give a potentially useful characterization of inessential automorphism groups for parabolic geometries.

math.DG↗