Celebratio Mathematica

[1] J. Simons: “On the transitivity of holonomy systems,” Ann. Math. (2) 76 : 2 (September 1962), pp. 213–234. Based on the author’s 1962 PhD thesis. MR 0148010 Zbl 0106.15201 article

Several years ago M. Berger [1955] gave a classification of possible candidates for the holonomy groups of manifolds having affine connections with zero torsion. His proofs depended strongly on the classification of simple Lie groups, and his technique consisted in eliminating various groups using the Bianchi identities and the theorem of Ambrose and Singer [1953].

The most striking of his results is the list he determines of possible holonomy groups of a riemannian manifold. These groups all turn out to be transitive on the unit sphere in the tangent space of the manifold, except in the case that the manifold is a symmetric space of rank \( \geq 2 \). It is natural to ask for an intrinsic proof of this rather startling fact, one which avoids the classification theorems.

Our object here is to give a purely algebraic generalization of the notion of a holonomy group, and to present in this setting a fairly short, intrinsic proof of the result on transitivity. Although we only treat that portion of the problem which has to do with riemannian manifolds, it is possible that the devices employed could be altered to pertain to other situations.

[2] J. H. Simons: On the transitivity of holonomy systems. Ph.D. thesis, UC-Berkeley, 1962. Advised by B. Kostant. An article based on this thesis was published in Ann. Math. 76:2 (1962). MR 2939297 phdthesis

[3] J. Simons: “Minimal cones, Plateau’s problem, and the Bernstein conjecture,” Proc. Natl. Acad. Sci. U.S.A. 58 : 2 (August 1967), pp. 410–411. MR 0216387 Zbl 0168.09903 article

The work of Fleming [1966], along with that of De Giorgi [1965], has shown that the Bernstein conjecture for graphs in \( \mathbb{R}^{n+1} \) would follow from an interior regularity theorem for \( (n-1) \)-dimensional minimal integral currents in \( \mathbb{R}^n \). The work of Federer and Fleming [1960], along with that of De Giorgi [1965] and Triscari [1963], has shown that such an interior regularity would follow from a theorem showing that the cone in \( \mathbb{R}^n \) over an \( (n-2) \)-dimensional, nontotally geodesic, closed minimal variety in \( \mathbb{S}^{n-1} \) is unstable with respect to its boundary.

Almgren [1966] showed that the cone over such a 2-dimensional variety in \( \mathbb{S}^3 \) is unstable, and this yielded the interior regularity theorem in \( \mathbb{R}^4 \) and the Bernstein conjecture in \( \mathbb{R}^5 \). In this note we announce an instability theorem which is valid for the cone over any such subvariety of \( \mathbb{S}^n \) for \( n < 6 \), and give an example of a cone over such a subvariety in \( \mathbb{S}^7 \) which is locally stable in the sense that every deformation initially increases area. These results yield the Bernstein conjecture through \( \mathbb{R}^8 \), interior regularity through \( \mathbb{R}^7 \), and a good candidate for a counterexample to interior regularity in \( \mathbb{R}^8 \).

[4] J. Simons: “A note on minimal varieties,” Bull. Am. Math. Soc. 73 : 3 (1967), pp. 491–495. MR 0208481 Zbl 0153.23201 article

In [1966] Almgren considered the situation of a closed minimal variety \( H \), of dimension 2 immersed in \( \mathbb{S}^3 \). He observed that the second fundamental form, a real valued bilinear form on the tangent space to \( H \), is in fact the real part of a holomorphic quadratic differential with respect to the conformal structure on \( H \) induced by the metric inherited from its immersion in \( \mathbb{S}^3 \). He used this fact to conclude that \( \mathbb{S}^2 \) could not be immersed as a minimal variety in \( \mathbb{S}^3 \) unless it was already totally geodesic. It turns out that under the most general circumstances the second fundamental form of a \( p \)-dim minimal subvariety of an \( n \)-dim Riemannian manifold satisfies a natural second-order elliptic differential equation which generalizes the holomorphic condition mentioned above. In the case that the ambient manifold is \( \mathbb{S}^n \) the equation may be used to show that a closed minimal subvariety of \( \mathbb{S}^n \), of arbitrary codimension, which does not twist too much is already totally geodesic. In a sense this theorem is analogous to Bernstein’s theorem for complete minimal subvarieties in \( \mathbb{R}^n \).

[5] J. Simons: “Minimal varieties in riemannian manifolds,” Ann. Math. (2) 88 : 1 (July 1968), pp. 62–105. Russian translations were published in Tselochislennyye potoki i minimal’nyye poverkhnosti (1973) and Matematika 16:6 (1972). MR 0233295 Zbl 0181.49702 article

Our object in this paper is twofold. First, we give a basic exposition of immersed minimal varieties in a riemannian manifold. The principal result of this general investigation is the derivation of the linear elliptic second order equation satisfied by the second fundamental form of any minimal variety in any ambient manifold.

Second, we apply these general results in a more detailed study of a minimal varieties in the sphere and in euclidean space. This study includes an estimation of a lower bound for the index and the nullity of a non-totally geodesic closed minimal variety immersed in \( \mathbb{S}^n \); a theorem which generalizes to arbitrary co-dimensions the theorem of De Giorgi [1965] concerning the image of the Gauss map of a closed co-dimension 1 variety in \( \mathbb{S}^n \); a theorem which estimates an upper bound for the minimum value taken by the scalar curvature of any closed non-totally geodesic minimal variety in \( \mathbb{S}^n \); a theorem which calculates an explicit neighborhood of the standard metric on \( \mathbb{S}^p \) isolating it in the space of metric obtained from non-totally geodesic minimal immersions in \( \mathbb{S}^n \); a theorem generalizing the result of Almgren [1966] in which we show that the cone over any co-dimension 1 non-totally geodesic closed minimal variety in \( \mathbb{S}^n \) is unstable with respect to its boundary for \( n\leq 6 \), and an example of a cone over such a variety in \( \mathbb{S}^7 \) which is at least locally stable. The consequences of this last result are an extension through dimension 7 of interior regularity of solutions to the co-dimension 1 Plateau problem, and an extension through dimension 8 of the Bernstein conjecture.

[6] S.-S. Chern and J. Simons: “Some cohomology classes in principal fiber bundles and their application to Riemannian geometry,” Proc. Natl. Acad. Sci. U.S.A. 68 : 4 (April 1971), pp. 791–794. MR 0279732 Zbl 0209.25401 article

[7] J. Simons: “Minimal varieties in riemannian manifolds,” Matematika 16 : 6 (1972), pp. 60–104. Russian translation of an article published in Ann. Math. 88:1 (1968). Zbl 0241.53034 article

[8] H. B. Lawson, Jr. and J. Simons: “On stable currents and their application to global problems in real and complex geometry,” Ann. Math. (2) 98 : 3 (November 1973), pp. 427–450. Dedicated to S. S. Chern on his 60th birthday. MR 0324529 Zbl 0283.53049 article

In this paper we use some techniques from the calculus of variations to study the topology and geometry of submanifolds of the sphere and the geometry of complex projective space.

The basic idea is to use the existence theorems of Federer and Fleming to represent integral (or \( \mathbb{Z}_p \)) homology classes by rectifiable currents of least mass, and then to show that under various geometric assumptions one can construct deformations of these currents which are mass decreasing. In the case of an immersed submanifold of \( \mathbb{S}^n \) we show that appropriate assumptions on its extrinsic geometry imply the vanishing of a given homology group. In the case of \( \mathbb{P}^n(\mathbb{C}) \) we show that the only stable integral currents are the formal sums of algebraic varieties.

H. Blaine Lawson, Jr.	Related
S. S. Chern	Related	Volume

[9] J. Simons: “Minimal varieties in riemannian manifolds,” pp. 132–202 in Tselochislennyye potoki i minimal’nyye poverkhnosti [Integral flows and minimal surfaces]. Mir, 1973. Russian translation of an article published in Ann. Math. 88:1 (1968). Zbl 0274.53057 incollection

[10] S.-S. Chern and J. Simons: “Characteristic forms and geometric invariants,” Ann. Math. (2) 99 : 1 (January 1974), pp. 48–69. MR 0353327 Zbl 0283.53036 article

[11] J.-P. Bourguignon, H. B. Lawson, and J. Simons: “Stability and gap phenomena for Yang–Mills fields,” Proc. Natl. Acad. Sci. U.S.A. 76 : 4 (April 1979), pp. 1550–1553. MR 526178 Zbl 0408.53023 article

It is shown that any weakly stable Yang–Mills field of type \( \mathit{SU}(2) \) or \( \mathit{SU}(3) \) on the four-sphere must be self-dual or anti-self-dual. Any Yang–Mills field on \( \mathbb{S}^n \), \( n \geq 5 \), is unstable. Examples of stable fields on \( \mathbb{S}^4 \) and \( \mathbb{S}^n/\Gamma \) for \( n \geq 5 \) and \( \Gamma \neq \{e\} \) are given. It is also shown that, for any Yang–Mills field \( R \) on \( \mathbb{S}^4 \), the pointwise condition \( \| R^-\|^2 < 3 \) (or \( \| R^+\|^2 < 3 \)) implies that \( R^- = 0 \) (or respectively that \( R^+ = 0 \)). In general, any Yang–Mills field \( R \) on \( \mathbb{S}^n \), \( n \geq 3 \), that satisfies the pointwise condition \( \| R\|^2 < \frac{1}{2}{2 \choose n} \) is trivial. If \( n = 3 \) or 4, the condition \( \| R\|^2 < \frac{1}{2}{2 \choose n} \) implies that either \( R \) is the trivial field or it is the direct sum of a trivial field with a field of tangent spinors carrying the standard connection.

H. Blaine Lawson, Jr.	Related
Jean Pierre Bourguignon	Related

[12] J. Cheeger and J. Simons: “Differential characters and geometric invariants,” pp. 50–80 in Geometry and topology (College Park, MD, 1983–1984). Edited by J. Alexander and J. Harer. Lecture Notes in Mathematics 1167. Springer (Berlin), 1985. MR 827262 Zbl 0621.57010 incollection

This paper first appeared in a collection of lecture notes which were distributed at the A.M.S. Summer Institute on Differential Geometry, held at Stanford in 1973. Since then it has been (and remains) the authors’ intention to make available a more detailed version. But, in the mean time, we continued to receive requests for the original notes. Moreover, the secondary invariants we discussed have recently arisen in some new contexts, e.g., in physics and in the work of Cheeger and Gromov on “collapse” (which was the subject of the first author’s lectures at the Special Year). For these reasons we decided to finally publish the notes, albeit in their original form.

Jeff Cheeger	Related
James W. Alexander	Related
John L. Harer	Related

[13] J. Simons: “My interaction with S. S. Chern,” pp. 176–178 in S. S. Chern: A great geometer of the twentieth century (Los Angeles, 1990). Edited by S.-T. Yau. International Press (Hong Kong), 1992. MR 1201357 incollection

S. S. Chern	Related	Volume
Shing-Tung Yau	Related

[14] J. Simons and D. Sullivan: “Structured bundles define differential K-theory,” pp. 1–3 in Géométrie différentielle, physique mathématique, mathématiques et société, I: Volume en l’honneur de Jean Pierre Bourguignon [Differential geometry, mathematical physics, mathematics and society, I: Volume in honor of Jean Pierre Bourguignon]. Edited by O. Hijazi. Astérisque 321–322. Société Mathématique de France (Paris), 2008. MR 2521641 Zbl 1204.57021 incollection

Oussama Hijazi	Related
Jean Pierre Bourguignon	Related
Dennis P. Sullivan	Related

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	AUTHOR = {Simons, James and Sullivan, Dennis},
	TITLE = {Structured bundles define differential
		 {K}-theory},
	BOOKTITLE = {G\'eom\'etrie diff\'erentielle, physique
		 math\'ematique, math\'ematiques et soci\'et\'e,
		 {I}: {V}olume en l'honneur de {J}ean
		 {P}ierre {B}ourguignon [Differential
		 geometry, mathematical physics, mathematics
		 and society, {I}: {V}olume in honor
		 of {J}ean {P}ierre {B}ourguignon]},
	EDITOR = {Hijazi, Oussama},
	SERIES = {Ast\'erisque},
	NUMBER = {321--322},
	PUBLISHER = {Soci\'et\'e Math\'ematique de France},
	ADDRESS = {Paris},
	YEAR = {2008},
	PAGES = {1--3},
	URL = {http://smf4.emath.fr/en/Publications/Asterisque/2008/321/html/smf_ast_321_1-3.php},
	NOTE = {MR:2521641. Zbl:1204.57021.},
	ISSN = {0303-1179},
	ISBN = {9782856292587},
}

[15] J. Simons and D. Sullivan: “Axiomatic characterization of ordinary differential cohomology,” J. Topol. 1 : 1 (2008), pp. 45–56. MR 2365651 Zbl 1163.57020 ArXiv math/0701077 article

The Cheeger–Simons differential characters, the Deligne cohomology in the smooth category, the Hopkins–Singer construction of ordinary differential cohomology, and the recent Harvey–Lawson constructions are each in two distinct ways abelian group extensions of known functors. In one description, these objects are extensions of integral cohomology by the quotient space of all differential forms by the subspace of closed forms with integral periods. In the other, they are extensions of closed differential forms with integral periods by the cohomology with coefficients in the circle. These two series of short-exact sequences mesh with two interlocking long-exact sequences (the Bockstein sequence and the de Rham sequence) to form a commutative DNA-like array of functors called the Character Diagram. Our first theorem shows that on the category of smooth manifolds and smooth maps, any package consisting of a functor into graded abelian groups together with four natural transformations that fit together so as to form a Character Diagram as mentioned earlier is unique up to a unique natural equivalence. Our second theorem shows that natural product structure on differential characters is uniquely characterized by its compatibility with the product structures on the known functors in the Character Diagram. The proof of our first theorem couples the naturality with results about approximating smooth singular cycles and homologies by embedded pseudomanifolds.

[16] J. Simons and D. Sullivan: “Structured vector bundles define differential K-theory,” pp. 579–599 in Quanta of maths: Conference on noncommutative geometry in honor of Alain Connes (Paris, 29 March–6 April 2007). Edited by E. Blanchard. Clay Mathematics Proceedings 11. American Mathematical Society (Providence, RI), 2010. MR 2732065 Zbl 1216.19009 incollection

Étienne Blanchard	Related
Dennis P. Sullivan	Related
Alain Connes	Related

[17] J. Simons and D. Sullivan: “The Atiyah–Singer index theorem and Chern–Weil forms,” pp. 643–645 in Special issue: In honor of Michael Atiyah and Isadore Singer, published as Pure Appl. Math. Q. 6 : 2 (2010). MR 2761861 Zbl 1222.58015 incollection

Isadore M. Singer	Related	Volume
M. F. Atiyah	Related	Volume
Dennis P. Sullivan	Related

[18] J. Simons and D. Sullivan: “Differential characters for K-theory,” pp. 353–361 in Metric and differential geometry: The Jeff Cheeger anniversary volume (Tianjin and Beijing, China, 11–15 May 2009). Edited by X. Dai and X. Rong. Progress in Mathematics 297. Springer (Berlin), 2012. MR 3220448 Zbl 1256.19007 incollection

We describe a sequence of results that begins with the introduction of differential characters on singular cycles in the seventies motivated by the search for invariants of geometry or more generally bundles with connections. The sequence passes through an Eilenberg–Steenrod type uniqueness result for ordinary differential cohomology using these characters and a construction of a differential K-theory using Grothendieck’s construction on classes of complex bundles with connection. The last element of the sequence returns full circle with a differential character definition of differential K-theory. The cycles in this definition of characters for differential K-theory are closed smooth manifolds provided with complex structures and hermitian connections on their stable tangent bundles.

Xiaochun Rong	Related
Dennis P. Sullivan	Related
Xianzhe Dai	Related
Jeff Cheeger	Related

James Simons

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