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Patrick J. Ryan

Publications and source records attributed to Patrick J. Ryan.

8 recordsLinked to original sources
arXiv2026

Cohomogeneity-One Ruled Hypersurfaces in $\mathbb{CP}^2$ and $\mathbb{C}H^2$

In this paper, we show how to construct a special class of ruled hypersurfaces in the nonflat complex space forms $\mathbb{CP}^n$ and $\mathbb{C}H^n$. This is done by taking an arbitrary smooth curve in a totally geodesic (complex) one-dimensional submanifold and erecting an orthogonal ruling over each of its points. Concentrating on the $n=2$ case, we also examine the special situation in which the base curve has constant geodesic curvature. We show that, in this case, the construction yields precisely the real-analytic hypersurfaces of cohomogeneity one that satisfy a certain transversality condition.

math.DG↗
arXiv2025

Tight Spherical Embeddings (Updated Version)

This is an updated version of a paper which appeared in the proceedings of the 1979 Berlin Colloquium on Global Differential Geometry. This paper contains the original exposition together with some notes by the authors made in 2025 (as indicated in the text) that give references to descriptions of progress made in the field since the time of the original version of the paper. The main result of this paper is that every compact isoparametric hypersurface $M^n \subset S^{n+1} \subset {\bf R}^{n+2}$ is tight, i.e., every non-degenerate linear height function $\ell_p$, $p \in S^{n+1}$, has the minimum number of critical points on $M^n$ required by the Morse inequalities. Since $M^n$ lies in the sphere $S^{n+1}$, this implies that $M^n$ is also taut in $S^{n+1}$, i.e., every non-degenerate spherical distance function has the minimum number of critical points on $M^n$. A second result is that the focal submanifolds of isoparametric hypersurfaces in $S^{n+1}$ must also be taut. The proofs of these results are based on M\"{u}nzner's fundamental work on the structure of a family of isoparametric hypersurfaces in a sphere.

math.DG↗
arXiv2024

On the Work of Cartan and M\"{u}nzner on Isoparametric Hypersurfaces

A hypersurface $M^n$ in a real space form ${\bf R}^{n+1}$, $S^{n+1}$, or $H^{n+1}$ is isoparametric if it has constant principal curvatures. This paper is a survey of the fundamental work of Cartan and M\"{u}nzner on the theory of isoparametric hypersurfaces in real space forms, in particular, spheres. This work is contained in four papers of Cartan published during the period 1938--1940, and two papers of M\"{u}nzner that were published in preprint form in the early 1970's, and as journal articles in 1980--1981. These papers of Cartan and M\"{u}nzner have been the foundation of the extensive field of isoparametric hypersurfaces, and they have all been recently translated into English by T. Cecil. The paper concludes with a brief survey of the recently completed classification of isoparametric hypersurfaces in spheres.

math.DG↗
arXiv2015

Hypersurfaces in $CP^2$ and $CH^2$ with two distinct principal curvatures

It is known that hypersurfaces in $CP^n$ or $CH^n$ for which the number $g$ of distinct principal curvatures satisfied $g \le 2$ must belong to a standard list of Hopf hypersurfaces with constant principal curvatures, provided that $n \ge 3$. In this paper, we construct a 2-parameter family of non-Hopf hypersurfaces in $CP^2$ and $CH^2$ with $g=2$ and show that every non-Hopf hypersurface with $g=2$ is locally of this form.

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arXiv2010

The *-Ricci tensor for hypersurfaces in CP^n and CH^n

We update and refine the work of T. Hamada concerning *-Einstein hypersurfaces in complex space forms CP^n and CH^n. We also address existence questions using the methods of moving frames and exterior differential systems.

math.DG↗
arXiv2008

The structure Jacobi operator for hypersurfaces in CP^2 and CH^2

Using the methods of moving frames, we study real hypersurfaces in complex projective space CP^2 and complex hyperbolic space CH^2 whose structure Jacobi operator has various special properties. Our results complement work of several other authors who worked on such hypersurfaces in CP^n and CH^n for n>2.

math.DG↗
arXiv2008

Hopf Hypersurfaces of Small Hopf Principal Curvature in CH^2

Using the methods of moving frames and exterior differential systems, we show that there exist Hopf hypersurfaces in complex hyperbolic space CH^2 with any specified value of the Hopf principal curvature less than or equal to the corresponding value for the horosphere. We give a construction for all such hypersurfaces in terms of Weierstrass-type data, and also obtain a classification of pseudo-Einstein hypersurfaces in CH^2.

math.DG↗