Horizontal Path Lifting for General Connections
We characterize the existence of horizontal path lifts for general connections on arbitrary fiber bundles with a new property that also gives fresh insight into linear and $G$-connections.
math.DG↗
arXiv subjects
Phillip E. Parker
Publications and source records attributed to Phillip E. Parker.
Nilsolitons of H-type in the Lorentzian setting
It is known that all left-invariant pseudo-Riemannian metrics on $H_3$ are algebraic Ricci solitons. We consider generalizations of Riemannian $H$-type, namely pseudo$H$-type and $pH$-type. We study algebraic Ricci solitons of left-invariant Lorentzian metrics on 2-step nilpotent Lie groups of both types.
math.DG↗
Lattices and Periodic Geodesics in Pseudoriemannian 2-step Nilpotent Lie Groups
We give a basic treatment of lattices $Γ$ in these groups. Certain tori $T_F$ and $T_B$ provide the model fiber and the base for a submersion of $Γ\backslash N$. This submersion may not be pseudoriemannian in the usual sense, because the tori may be degenerate. We then begin the study of periodic geodesics in these compact nilmanifolds, obtaining a complete calculation of the period spectrum of certain flat spaces.
math.DG↗
Pseudoriemannian Nilpotent Lie Groups
This is a survey article with a limited list of references (as required by the publisher) which appears in the Encyclopedia of Mathematical Physics, eds. J.-P. Francoise, G.L. Naber and Tsou S.T. Oxford: Elsevier, 2006. vol.4, pp.94--104.
math.DG↗
Conjugate loci of pseudoriemannian 2-step nilpotent Lie groups with nondegenerate center
We determine the complete conjugate locus along all geodesics parallel or perpendicular to the center (Theorem 2.3). When the center is 1-dimensional we obtain formulas in all cases (Theorem 2.5), and when a certain operator is also diagonalizable these formulas become completely explicit (Corollary 2.7). These yield some new information about the smoothness of the pseudoriemannian conjugate locus. We also obtain the multiplicities of all conjugate points.
math.DG↗
Skewadjoint operators on pseudoeuclidean spaces
We give a complete classification in canonical forms on finite-dimensional vector spaces over the real numbers.
math.AC↗
Pseudoriemannian 2-Step Nilpotent Lie Groups
We begin a systematic study of these spaces, initially following along the lines of Eberlein's comprehensive study of the Riemannian case. In particular, we integrate the geodesic equation, discuss the structure of the isometry group, and make a study of lattices and periodic geodesics. Some major differences from the Riemannian theory appear. There are many flat groups (versus none), including Heisenberg groups. While still a semidirect product, the isometry group can be strictly larger than the obvious analogue. Everything is illustrated with explicit examples. We introduce the notion of pH-type, which refines Kaplan's H-type and completes Ciatti's partial extension. We give a general construction for algebras of pH-type.
math.DG↗