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Robert L. Bryant

Publications and source records attributed to Robert L. Bryant.

At least 19 recordsLinked to original sources
arXiv2025

On the conformal volume of 2-tori

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 not covered by the conjecture of Montiel and Ros. Originally, I did not think that this claim was of enough interest to warrant including the (somewhat involved) proof, but time has shown otherwise. (Also, the proof included here is shorter than my original proof; instead, it relies on a MAPLE computation.)

math.DG↗
arXiv2025

Some remarks on G_2-structures

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consists of some loosely related remarks about the geometry of G_2-structures on 7-manifolds and is partly based on old unpublished joint work with two other people: F. Reese Harvey and Steven Altschuler.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 subsumed in the work of Hitchin \cite{MR02m:53070} and Joyce \cite{MR01k:53093}. I am making it available now mainly because of interest expressed by others in seeing these results written up since they do not seem to have all made it into the literature. A formula is derived for the scalar curvature and Ricci curvature of a G_2-structure in terms of its torsion. When the fundamental 3-form of the G_2-structure is closed, this formula implies, in particular, that the scalar curvature of the underlying metric is nonpositive and vanishes if and only if the structure is torsion-free. This version contains some new results on the pinching of Ricci curvature for metrics associated to closed G_2-structures. Some formulae are derived for closed solutions of the Laplacian flow that specify how various related quantities, such as the torsion and the metric, evolve with the flow. These may be useful in studying convergence or long-time existence for given initial data.

math.DG↗
arXiv2024

Hessianizability of surface metrics

A symmetric quadratic form $g$ on a surface~$M$ is said to be locally Hessianizable if each $p\in M$ has an open neighborhood~$U$ on which there exists a local coordinate chart $(x^1,x^2):U\to\mathbb{R}^2$ and a function $f:U\to\mathbb{R}$ such that, on $U$, we have $$ g = \frac{\partial^2 f}{\partial x^i\partial x^j}\,\mathrm{d} x^i\circ\mathrm{d} x^j. $$ In this article, I show that, when $g$ is nondegenerate and smooth, it is always smoothly locally Hessianizable.

math.DG↗
arXiv2023

On the Convex Pfaff-Darboux Theorem of Ekeland and Nirenberg

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 forms needed in these models often come with an additional requirement of some type of convexity, which is not provided by the classical proofs of the Pfaff-Darboux theorem. (The appropriate notion of 'convexity' is a feature of the economic model. In the simplest case, when the economic model is formulated in a domain in $\mathbb{R}^n$, convexity has its usual meaning.) In [Methods Appl. Anal. 9 (2002), 329-344], Ekeland and Nirenberg were able to characterize necessary and sufficient conditions for a given 1-form $ω$ to admit a convex local normal form (and to show that some earlier attempts [Chiappori P.-A., Ekeland I., Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4 25 (1997), 287-297] and [Zakalyukin V.M., C. R. Acad. Sci. Paris Sér. I Math. 327 (1998), 633-638] at this characterization had been unsuccessful). In this article, after providing some necessary background, I prove a strengthened and generalized convex Pfaff-Darboux theorem, one that covers the case of a Legendrian foliation in which the notion of convexity is defined in terms of a torsion-free affine connection on the underlying manifold. (The main result of Ekeland and Nirenberg concerns the case in which the affine connection is flat.)

math.DG↗
arXiv2023

The generality of closed $G_2$ solitons

The local generality of the space of solitons for the Laplacian flow of closed $G_2$-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 E. Cartan). The method is to construct a natural exterior differential system whose integral manifolds describe such solitons and to show that it is involutive in Cartan's sense, so that Cartan-Kahler theory can be applied. Meanwhile, it turns out that, for the more special case of gradient solitons, the natural exterior differential system is not involutive, and the generality of these structures remains a mystery.

math.DG↗
arXiv2021

S.-S. Chern's study of almost-complex structures on the six-sphere

In 2003, S.-s. Chern began a study of almost-complex structures on the 6-sphere, with the idea of exploiting the special properties of its well-known almost-complex structure invariant under the exceptional group $G_2$. While he did not solve the (currently still open) problem of determining whether there exists an integrable almost-complex structure on the 6-sphere, he did prove a significant identity that resolves the question for an interesting class of almost-complex structures on the 6-sphere.

math.DG↗
arXiv2020

Notes on spinors in low dimension

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), (9,0), (9,1), (10,0), (10,1), and (10,2). I'm making them available on the arXiv because I continue to get requests for them as well as questions about how they can be cited.

math.GR↗
arXiv2020

A circle quotient of a $G_2$ cone

A study is made of $R^6$ as a singular quotient of the conical space $R^+\times CP^3$ with holonomy $G_2$ with respect to an obvious action by $U(1)$ on $CP^3$ with fixed points. Closed expressions are found for the induced metric, and for both the curvature and symplectic 2-forms characterizing the reduction. All these tensors are invariant by a diagonal action of $SO(3)$ on $R^6$, which can be used effectively to describe the resulting geometrical features.

math.DG↗
arXiv2020

Flat Metrics with a Prescribed Derived Coframing

The following problem is addressed: A $3$-manifold $M$ is endowed with a triple $Ω= \big(Ω^1,Ω^2,Ω^3\big)$ of closed $2$-forms. One wants to construct a coframing $ω= \big(ω^1,ω^2,ω^3\big)$ of $M$ such that, first, ${\rm d}ω^i = Ω^i$ for $i=1,2,3$, and, second, the Riemannian metric $g=\big(ω^1\big)^2+\big(ω^2\big)^2+\big(ω^3\big)^2$ be flat. We show that, in the 'nonsingular case', i.e., when the three $2$-forms $Ω^i_p$ span at least a $2$-dimensional subspace of $Λ^2(T^*_pM)$ and are real-analytic in some $p$-centered coordinates, this problem is always solvable on a neighborhood of $p\in M$, with the general solution $ω$ depending on three arbitrary functions of two variables. Moreover, the characteristic variety of the generic solution $ω$ can be taken to be a nonsingular cubic. Some singular situations are considered as well. In particular, we show that the problem is solvable locally when $Ω^1$, $Ω^2$, $Ω^3$ are scalar multiples of a single 2-form that do not vanish simultaneously and satisfy a nondegeneracy condition. We also show by example that solutions may fail to exist when these conditions are not satisfied.

math.DG↗
arXiv2019

Notes on Projective, Contact, and Null Curves

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 this note consists of recounting the classical background. The main application is the explicit classification of rational null curves of low degree in Q^3. I have recently received a number of requests for these notes, so I am posting them to make them generally available.

math.AG↗
arXiv2014

Notes on exterior differential systems

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.

math.DG↗
arXiv2012

Nonembedding and nonextension results in special holonomy

Constructions of metrics with special holonomy by methods of exterior differential systems are reviewed and the interpretations of these construction as `flows' on hypersurface geometries are considered. It is shown that these hypersurface 'flows' are not generally well-posed for smooth initial data and counterexamples to existence are constructed.

math.DG↗
arXiv2012

Some differential complexes within and beyond parabolic geometry

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 a parabolic geometry, our complexes coincide with the Bernstein-Gelfand-Gelfand complex associated with the trivial representation. However, at least in the cases we discuss, our constructions are relatively simple and avoid most of the machinery of parabolic geometry. Moreover, our method extends to certain geometries beyond the parabolic realm.

math.DG↗
arXiv2011

Complex analysis and a class of Weingarten surfaces

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 f, where H and K are the mean and Gauss curvatures, respectively. The results are either not new or are minor extensions of known results, but the method, which involves introducing a different conformal structure on the surface than the one induced by the first fundamental form, is different from the one used by Hopf and requires less technical results from the theory of PDE than Hopf's methods. This is a TeXed version of a manuscript dating from early 1984. It was never submitted for publication, though it circulated to some people and has been referred to from time to time in published articles. It is being provided now for the convenience of those who have asked for a copy. Except for the correction of various grammatical or typographical mistakes and infelicities and the addition of some (clearly marked) comments at the end of the introduction, the text is that of the original.

math.DG↗
arXiv2010

Metrisability of two-dimensional projective structures

We carry out the programme of R. Liouville \cite{Liouville} 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 a projective class. It can be expressed as a point invariant for a second order ODE whose integral curves are the geodesics of $[Γ]$ or as a weighted scalar projective invariant of the projective class. If the obstruction vanishes we find the sufficient conditions for the existence of a metric in the real analytic case. In the generic case they are expressed by the vanishing of two invariants of order 6 in the connection. In degenerate cases the sufficient obstruction is of order at most 8.

math.DG↗
arXiv2005

Conformal geometry and 3-plane fields on 6-manifolds

The purpose of this note is to provide yet another example of the link between certain conformal geometries and ordinary differential equations, along the lines of the examples discussed by Nurowski in math.DG/0406400. In this particular case, I consider the equivalence problem for 3-plane fields D on 6-manifolds M that satisfy the nondegeneracy condition that D+[D,D]=TM I give a solution of the equivalence problem for such D (as Tanaka has previously), showing that it defines a so(4,3)-valued Cartan connection on a principal right H-bundle over M where H is the subgroup of SO(4,3) that stabilizes a null 3-plane in R^{4,3}. Along the way, I observe that there is associated to each such D a canonical conformal structure of split type on M, one that depends on two derivatives of the plane field D. I show how the primary curvature tensor of the Cartan connection associated to the equivalence problem for D can be interpreted as the Weyl curvature of the associated conformal structure and, moreover, show that the split conformal structures in dimension~6 that arise in this fashion are exactly the ones whose so(4,4)-valued Cartan connection admits a reduction to a spin(4,3)-connection. I also discuss how this case has features that are analogous to those of Nurowski's examples.

math.DG↗
arXiv2005

Remarks on the geometry of almost complex 6-manifolds

This article is mostly a writeup of two talks, the first given in the Besse Seminar at the Ecole Polytechnique in 1998 and the second given at the 2000 International Congress on Differential Geometry in memory of Alfred Gray in Bilbao, Spain. It begins with a discussion of basic geometry of almost complex 6-manifolds. In particular, I define a 2-parameter family of intrinsic first-order functionals on almost complex structures on 6-manifolds and compute their Euler-Lagrange equations. It also includes a discussion of a natural generalization of holomorphic bundles over complex manifolds to the almost complex case. The general almost complex manifold will not admit any nontrivial bundles of this type, but there is a large class of nonintegrable almost complex manifolds for which there are such nontrivial bundles. For example, the standard almost complex structure on the 6-sphere admits such nontrivial bundles. This class of almost complex manifolds in dimension 6 will be referred to as quasi-integrable. Some of the properties of quasi-integrable structures (both almost complex and unitary) are developed and some examples are given. However, it turns out that quasi-integrability is not an involutive condition, so the full generality of these structures in Cartan's sense is not well-understood.

math.DG↗