Celebratio Mathematica

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.

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.

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

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

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James Simons

James Harris Simons

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