Skip to main content

Research Repository

Advanced Search

Outputs (31)

Regularity and Continuity properties of the sub-Riemannian exponential map (2023)
Journal Article
Borza, S., & Klingenberg, W. (2023). Regularity and Continuity properties of the sub-Riemannian exponential map. Journal of Dynamical and Control Systems, 29(4), 1385-1407. https://doi.org/10.1007/s10883-022-09624-y

We prove a version of Warner’s regularity and continuity properties for the sub-Riemannian exponential map. The regularity property is established by considering sub-Riemannian Jacobi fields while the continuity property follows from studying the Mas... Read More about Regularity and Continuity properties of the sub-Riemannian exponential map.

Evolving to Non-round Weingarten Spheres: Integer Linear Hopf Flows (2021)
Journal Article
Guilfoyle, B., & Klingenberg, W. (2021). Evolving to Non-round Weingarten Spheres: Integer Linear Hopf Flows. Partial Differential Equations and Applications, 2(6), Article 72. https://doi.org/10.1007/s42985-021-00128-1

In the 1950’s Hopf gave examples of non-round convex 2-spheres in Euclidean 3-space with rotational symmetry that satisfy a linear relationship between their principal curvatures. In this paper, we investigate conditions under which evolving a smooth... Read More about Evolving to Non-round Weingarten Spheres: Integer Linear Hopf Flows.

Fredholm-regularity of holomorphic discs in plane bundles over compact surfaces (2020)
Journal Article
Guilfoyle, B., & Klingenberg, W. (2020). Fredholm-regularity of holomorphic discs in plane bundles over compact surfaces. Annales de la Faculté des sciences de Toulouse (En ligne), 29(3), 565-576. https://doi.org/10.5802/afst.1639

We study the space of holomorphic discs with boundary on a surface in a real 2-dimensional vector bundle over a compact 2-manifold. We prove that, if the ambient 4-manifold admits a fibre-preserving transitive holomorphic action, then a section with... Read More about Fredholm-regularity of holomorphic discs in plane bundles over compact surfaces.

A global version of a classical result of Joachimsthal (2019)
Journal Article
Guilfoyle, B., & Klingenberg, W. (2019). A global version of a classical result of Joachimsthal. Houston journal of mathematics, 45(2), 455-467

A classical result attributed to Joachimsthal in 1846 states that if two surfaces intersect with constant angle along a line of curvature of one surface, then the curve of intersection is also a line of curvature of the other surface. In this note we... Read More about A global version of a classical result of Joachimsthal.

Mean Curvature Flow of Compact Spacelike Submanifolds in Higher Codimension (2019)
Journal Article
Guilfoyle, B., & Klingenberg, K. (2019). Mean Curvature Flow of Compact Spacelike Submanifolds in Higher Codimension. Transactions of the American Mathematical Society, 372(9), 6263-6281. https://doi.org/10.1090/tran/7766

We prove the longtime existence for mean curvature flow of a smooth n-dimensional spacelike submanifold of an (n + m)-dimensional manifold whose metric satisfies the timelike curvature condition.

Parabolic Classical Curvature Flows (2017)
Journal Article
Guilfoyle, B., & Klingenberg, W. (2018). Parabolic Classical Curvature Flows. Journal of the Australian Mathematical Society, 104(3), 338-357. https://doi.org/10.1017/s1446788717000210

We consider classical curvature flows: 1-parameter families of convex embeddings of the 2-sphere into Euclidean 3-space, which evolve by an arbitrary (nonhomogeneous) function of the radii of curvature (RoC). We determine conditions for parabolic flo... Read More about Parabolic Classical Curvature Flows.

A Converging Lagrangian Flow in the Space of Oriented Line (2016)
Journal Article
Guilfoyle, B., & Klingenberg, W. (2016). A Converging Lagrangian Flow in the Space of Oriented Line. Kyushu journal of mathematics, 70(2), 343-351. https://doi.org/10.2206/kyushujm.70.343

Under mean radius of curvature flow, a closed convex surface in Euclidean space is known to expand exponentially to infinity. In the three-dimensional case we prove that the oriented normals to the flowing surface converge to the oriented normals of... Read More about A Converging Lagrangian Flow in the Space of Oriented Line.

Totally null surfaces in neutral Kähler 4-manifolds (2016)
Journal Article
Georgiou, N., Guilfoyle, B., & Klingenberg, W. (2016). Totally null surfaces in neutral Kähler 4-manifolds. Balkan Journal of Geometry and its Applications, 21(1), 27-41

We study the totally null surfaces of the neutral Kähler metric on certain 4-manifolds. The tangent spaces of totally null surfaces are either self-dual (α-planes) or anti-self-dual (β-planes) and so we consider α-surfaces and β-surfaces. The metric... Read More about Totally null surfaces in neutral Kähler 4-manifolds.

On the geometry of spaces of oriented geodesics. (2011)
Journal Article
Alekseevsky, D., Guilfoyle, B., & Klingenberg, W. (2011). On the geometry of spaces of oriented geodesics. Annals of Global Analysis and Geometry, 40(4), 389-409. https://doi.org/10.1007/s10455-011-9261-5

Let M be either a simply connected pseudo-Riemannian space of constant curvature or a rank one Riemannian symmetric space, and consider the space L(M) of oriented geodesics of M. The space L(M) is a smooth homogeneous manifold and in this paper we de... Read More about On the geometry of spaces of oriented geodesics..

On Weingarten surfaces in Euclidean and Lorentzian 3-space (2010)
Journal Article
Guilfoyle, B., & Klingenberg, W. (2010). On Weingarten surfaces in Euclidean and Lorentzian 3-space. Differential Geometry and its Applications, 28(4), 454-468. https://doi.org/10.1016/j.difgeo.2009.12.002

We study the neutral Kähler metric on the space of time-like lines in Lorentzian View the MathML source, which we identify with the total space of the tangent bundle to the hyperbolic plane. We find all of the infinitesimal isometries of this metric,... Read More about On Weingarten surfaces in Euclidean and Lorentzian 3-space.

On C2-smooth Surfaces of Constant Width. (2009)
Journal Article
Guilfoyle, B., & Klingenberg, W. (2009). On C2-smooth Surfaces of Constant Width. Tbilisi Mathematical Journal, 2, 1-17

In this paper, we obtain a number of results for C2-smooth surfaces of constant width in Euclidean 3-space E3-. In particular, we establish an integral inequality for constant width surfaces. This is used to prove that the ratio of volume to cubed wi... Read More about On C2-smooth Surfaces of Constant Width..

Geodesic Flow on Global Holomorphic Sections of TS^2 (2007)
Journal Article
Klingenberg, W., & Guilfoyle, B. (2007). Geodesic Flow on Global Holomorphic Sections of TS^2. Bulletin of the Belgian Mathematical Society Simon Stevin (Printed), 14(2), 363-371

We study the geodesic flow on the global holomorphic sections of the bundle π:TS2→S2π:TS2→S2 induced by the neutral Kähler metric on the space of oriented lines of R3R3, which we identify with TS2TS2. This flow is shown to be completely integrable wh... Read More about Geodesic Flow on Global Holomorphic Sections of TS^2.

Isolated umbilic points on surfaces in R^3 (2006)
Journal Article
Klingenberg, W., & Guilfoyle, B. (2006). Isolated umbilic points on surfaces in R^3. Deltio tīs Ellīnikīs mathīmatikīs etaireias (1960), 51, 23-30

Reflection of a wave off a surface. (2006)
Journal Article
Klingenberg, W., & Guilfoyle, B. (2006). Reflection of a wave off a surface. Journal of Geometry, 84(1), 55-72. https://doi.org/10.1007/s00022-005-0022-0

Recent investigations of the space of oriented lines in R3R3 are applied to geometric optics. The general formulae for reflection of a wavefront in a surface are derived and in three special cases explicit descriptions are provided: when the reflecti... Read More about Reflection of a wave off a surface..

The Casimir effect between non-parallel plates by geometric optics (2005)
Journal Article
Guilfoyle, B., Klingenberg, W., & Sen, S. (2005). The Casimir effect between non-parallel plates by geometric optics. Reviews in Mathematical Physics, 17(8), 859 - 880. https://doi.org/10.1142/s0129055x05002431

The first two authors have developed a technique which uses the complex geometry of the space of oriented affine lines in ℝ3 to describe the reflection of rays off a surface. This can be viewed as a parametric approach to geometric optics which has m... Read More about The Casimir effect between non-parallel plates by geometric optics.

An indefinite Kaehler metric on the space of oriented lines (2005)
Journal Article
Guilfoyle, B., & Klingenberg, W. (2005). An indefinite Kaehler metric on the space of oriented lines. Journal of the London Mathematical Society, 72(2), 497-509. https://doi.org/10.1112/s0024610705006605

The total space of the tangent bundle of a Kähler manifold admits a canonical Kähler structure. Parallel translation identifies the space ${\mathbb{T}}$ of oriented affine lines in ${\mathbb{R}}^3$ with the tangent bundle of $S^2$. Thus the round met... Read More about An indefinite Kaehler metric on the space of oriented lines.

Generalized surfaces in R^3 (2004)
Journal Article
Klingenberg, W., & Guilfoyle, B. (2004). Generalized surfaces in R^3. Mathematical Proceedings of the Royal Irish Academy, 104A(2), 199-209

On the space of oriented affine lines in R^3 (2004)
Journal Article
Guilfoyle, B., & Klingenberg, W. (2004). On the space of oriented affine lines in R^3. Archiv der Mathematik, 82(1), 81 - 84. https://doi.org/10.1007/s00013-003-4861-3

We introduce a local coordinate description for the correspondence between the space of oriented affine lines in Euclidean and the tangent bundle to the 2-sphere. These can be utilised to give canonical coordinates on surfaces in, as we illustrate wi... Read More about On the space of oriented affine lines in R^3.

Real hypersurfaces of Kahler manifolds (2001)
Journal Article
Klingenberg, W. (2001). Real hypersurfaces of Kahler manifolds. Asian Journal of Mathematics, 5(1), 1 -18. https://doi.org/10.4310/ajm.2001.v5.n1.a1

Building on work by S. M. Webster \ref[J. Differential Geom. 13 (1978), no. 1, 25--41; MR0520599 (80e:32015)] the author studies the geometry of the second fundamental form of a real hypersurface in a Kähler manifold. As an application he proves that... Read More about Real hypersurfaces of Kahler manifolds.

Moduli space of holomorphic discs with boundary. (1997)
Journal Article
Klingenberg, W. (1997). Moduli space of holomorphic discs with boundary. Archiv der Mathematik, 69(3), 249-253. https://doi.org/10.1007/s000130050117

We study the moduli problem for holomorphic discs with boundary constrained to be in a maximally real submanifold and establish smooth dependence of the associated moduli space on the boundary condition.