Journal articleAdvances in Mathematics · May 14, 2025
We provide a classification of curvature homogeneous hypersurfaces in space forms by classifying the ones in and . In higher dimensions, besides the isoparametric and the constant curvature ones, there is a single one in . Besides the obvious examples, we ...
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Journal articlePure and Applied Mathematics Quarterly · January 31, 2024
The local generality of the space of solitons for the Laplacian flow of closed G2-structures is analyzed, and it is shown that the germs of such structures depend, up to diffeomorphism, on 16 functions of 6 variables (in the sense of É. Cartan). The method ...
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Journal articleInternational Journal of Mathematics · November 1, 2021
In April 2003, Chern began a study of almost-complex structures on the six-sphere, with the idea of exploiting the special properties of its well-known almost-complex structure invariant under the exceptional group G2. While he did not solve the ...
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Journal articleJournal of Differential Geometry · January 1, 2021
We study non-reversible Finsler metrics with constant flag curvature 1 on S2 and show that the geodesic flow of every such metric is conjugate to that of one of Katok's examples, which form a 1- parameter family. In particular, the length of the shortest c ...
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Journal articleDifferential Geometry and Its Application · December 1, 2020
A study is made of R6 as a singular quotient of the conical space R+×CP3 with holonomy G2, with respect to an obvious action by U(1) on CP3 with fixed points. Closed expressions are found for the induc ...
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Journal article · November 9, 2020
The purpose of these old notes (written in 1998 during a research project on holonomy of pseudo-Riemannian manifolds of type (10,1)) is to determine the orbit structure of the groups Spin(p,q) acting on their spinor spaces for the values (p,q) = (8,0), (9, ...
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Journal articleSymmetry Integrability and Geometry Methods and Applications Sigma · January 1, 2020
The following problem is addressed: A 3-manifold M is endowed with a triple Ω =(Ω1, Ω2, Ω3) of closed 2-forms. One wants to construct a coframing ω =(ω1, ω2, ω3) of M such that, first, dω
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Journal articleAdvanced Studies in Pure Mathematics · November 22, 2019
For smooth manifolds equipped with various geometric structures, we construct
complexes that replace the de Rham complex in providing an alternative fine
resolution of the sheaf of locally constant functions. In case that the
geometric structure is that of ...
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Journal article · May 15, 2019
These are notes on some algebraic geometry of complex projective curves, together with an application to studying the contact curves in CP^3 and the null curves in the complex quadric Q^3 in CP^4, related by the well-known Klein correspondence. Most of thi ...
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Journal articleJournal of Geometry and Physics · June 1, 2017
We study two-dimensional Finsler metrics of constant flag curvature and show that such Finsler metrics that admit a Killing field can be written in a normal form that depends on two arbitrary functions of one variable. Furthermore, we find an approach to c ...
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Journal articleSIGMA 19 · December 22, 2015
The classical Pfaff-Darboux theorem, which provides local 'normal forms' for $1$-forms on manifolds, has applications in the theory of certain economic models [Chiappori P.-A., Ekeland I., Found. Trends Microecon. 5 (2009), 1-151]. However, the normal form ...
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Journal article · July 6, 2015
This note (originally from 2015) provides a proof of a 1985 conjecture of Montiel and Ros concerning the conformal volume of tori. This updated version adds a proof of the claim made in Remark 5 about the value of the conformal volume of tori in the cases ...
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Journal article · May 13, 2014
These are notes for a very rapid introduction to the basics of exterior differential systems and their connection with what is now known as Lie theory, together with some typical and not-so-typical applications to illustrate their use. ...
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Journal article · January 10, 2011
We prove short time existence and uniqueness of solutions to the Laplacian flow for closed $G_2$ structures on a compact manifold $M^7$. The result was claimed in \cite{BryantG2}, but its proof has never appeared. ...
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Journal articleJournal of Differential Geometry · January 1, 2009
We carry out the programme of R. Liouville [19] to construct an explicit local obstruction to the existence of a Levi-Civita connection within a given projective structure [Γ] on a surface. The obstruction is of order 5 in the components of a connection in ...
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Journal articleMathematische Annalen · 2008
We give a complete list of normal forms for the two-dimensional metrics that admit a transitive Lie pseudogroup of geodesic-preserving transformations and we show that these normal forms are mutually non-isometric. This solves a problem posed by Sophus Lie ...
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Journal article · July 27, 2004
A unimodular complex surface is a complex 2-manifold X endowed with a holomorphic volume form. A strictly pseudoconvex real hypersurface M in X inherits not only a CR-structure but a canonical coframing as well.
In this article, this canonical coframing ...
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Journal articleDiscrete and Computational Geometry · January 1, 2004
The motion of a biomolecule greatly depends on the engulfing solution, which is mostly water. Instead of representing individual water molecules, it is desirable to develop implicit solvent models that nevertheless accurately represent the contribution of ...
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Journal article · June 24, 2000
I use local differential geometric techniques to prove that the algebraic cycles in certain extremal homology classes in Hermitian symmetric spaces are either rigid (i.e., deformable only by ambient motions) or quasi-rigid (roughly speaking, foliated by ri ...
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Journal articleSeminaire Bourbaki · October 11, 1999
This article is a report on the status of the problem of classifying the irriducibly acting subgroups of GL(n,R) that can appear as the holonomy of a torsion-free affine connection. In particular, it contains an account of the completion of the classificat ...
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Journal articleAdv. Stud. Pure Math., 37, Math. Soc. Japan, Tokyo, 2002, 1--44 · September 27, 1999
The purpose of this article is to classify the real hypersurfaces in complex space forms of dimension 2 that are both Levi-flat and minimal. The main results are as follows:
When the curvature of the complex space form is nonzero, there is a 1-parameter ...
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Journal articlePhysics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics · 1999
It was recently pointed out by E. Witten that for a D-brane
to consistently wrap a submanifold of some manifold, the
normal bundle must admit a Spin^c structure. We examine
this
constraint in the case of type II string compactifications
with vanishing ...
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Journal articleBoletim Da Sociedade Brasileira De Matematica · September 1, 1991
We prove several structure theorems about the special class of minimal submanifolds which Harvey and Lawson have called "austere" and which arose in connection with their foundational work on calibrations. The condition of austerity is a pontwise condition ...
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Journal articleTransactions of the American Mathematical Society · January 1, 1985
In this note, we study an overdetermined system of partial differential equations whose solutions determine the minimal surfaces in Sn of constant Gaussian curvature. If the Gaussian curvature is positive, the solution to the global problem was found by [C ...
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Journal articleTransactions of the American Mathematical Society · January 1, 1982
A CR-manifold is said to be Lorentzian if its Levi form has one negative eigenvalue and the rest positive. In this case, it is possible that the CR-manifold contains holomorphic curves. In this paper, necessary and sufficient conditions are derived (in ter ...
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https://orcid.org/0000-0002-4890-2471