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 (currently ...
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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 induced metric, and for both the curvature and symplectic 2- ...
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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, dωi = Ωi for i = 1, 2, 3, and, second, the Riemannian metric g = (ω1)2 ...
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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 provides a proof of a 1985 conjecture of Montiel and Ros about the
conformal volume of tori. (This material is not really new; I'm making it
available now because of requests related to recent interest in the
conjecture.) ...
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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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Other · May 27, 2011
An idea of Hopf's for applying complex analysis to the study of constant mean
curvature spheres is generalized to cover a wider class of spheres, namely,
those satisfying a Weingarten relation of a certain type, namely H = f(H^2-K)
for some smooth function ...
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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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Book · November 15, 2006
The DD6 Symposium was, like its predecessors DD1 to DD5 both a research symposium and a summer seminar and concentrated on differential geometry. This volume contains a selection of the invited papers and some additional contributions. ...
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Book · November 1, 2004
These expository accounts treat issues related to volume, geodesics, curvature and mathematical biology, with instructive examples. ...
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Book · November 1, 2004
These expository accounts treat issues related to volume, geodesics, curvature and mathematical biology, with instructive examples. ...
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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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Chapter · May 8, 2003
This article consists of some loosely related remarks about the geometry of
G_2-structures on 7-manifolds and is partly based on old unpublished joint work
with two other people: F. Reese Harvey and Steven Altschuler. Much of this work
has since been subsu ...
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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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Chapter · April 11, 2000
I discuss geometry and normal forms for pseudo-Riemannian metrics with
parallel spinor fields in some interesting dimensions. I also discuss the
interaction of these conditions for parallel spinor fields with the condition
that the Ricci tensor vanish (whi ...
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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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Chapter · June 23, 1994
Consider two manifolds~$M^m$ and $N^n$ and a first-order Lagrangian $L(u)$
for mappings $u:M\to N$, i.e., $L$ is an expression involving $u$ and its first
derivatives whose value is an $m$-form (or more generally, an $m$-density)
on~$M$. One is usually int ...
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Journal ArticleBoletim da Sociedade Brasileira de Matemática · 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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Chapter · 1988
A survey paper. However, there are some new results. Building on the results in A duality theorm for Willmore surfaces, I use the Klein correspondance to determine the moduli space of Willmore critical spheres for low critical values and also determine the ...
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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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