Elie Cartan made fundamental contributions to the theory of Lie groups and their geometrical applications. Among those, we can list the introduction of the remarkable family of Riemannian symmetric spaces, and his four papers that revolutionized the theory of isoparametric hypersurfaces.

How come Cartan did not notice the close relationship between symmetric spaces and isoparametric hypersurfaces?

Starting in 1926, Cartan developed his theory of symmetric spaces and published papers between 1927 and 1935. He first introduced them as Riemannian manifolds with parallel curvature tensor, under the name "espaces $\mathcal E$". Then he noticed that an equivalent, more geometric definition is to require that the geodesic symmetric around any point is an isometry and, around 1929, changed their name to "espaces symetriques'". The second definition implies that a symmetric space is a homogeneous space $G/K$ and there is a decompositon $\mathfrak g=\mathfrak k+\mathfrak p$ into the eigenspaces of an involution induced by conjugation by the geodesic symmetry at the basepoint. The adjoint action of $K$ on $\mathfrak p$ is equivalent to the linear isotropy representation of the symmetric space. The rank of $G/K$ is the dimension of a maximal flat, and coincides with the codimension of the principal orbits of this representation.

Isoparametric hypersurfaces in space forms are hypersurfaces with the simplest local invariants, namely, they have constant principal curvatures. They existed before Cartan, but between 1937 and 1940 he published four papers that completely revolutionized the field. Among other things, he showed that isoparametric hypersurfaces in spheres is a much more interesting subject than in Euclidean or hyperbolic spaces. Denote by $g$ the (constant) number of principal curvatures. The initial cases are not very interesting; in a sphere $S^{n+1}$, an isoparametric hypersurface with $g=1$ is an umbilic sphere, and with $g=2$ is the standard product of two spheres. Cartan showed that in case $g=3$ there are exactly four examples, of dimension $n=3d$ where $d=1$, $2$, $4$ or $8$ is the uniform multiplicity of the principal curvatures, each related to an embedding of a projective plane over one the normed division algebras $\mathbb R$, $\mathbb C$, $\mathbb H$, $\mathbb O$. He notes that those examples are all homogeneous and determines their isometry groups; in particular, he is pleased with the appearance of the exceptional group $F_4$ the case $n=24$, "(...) the first appearance of the simple $52$-dimensional group in a geometric problem (...)"; this group had already appeared in his classification of symmetric spaces. Later Cartan discusses the case $g=4$ and shows there are only two examples where the multiplicities of principal curvatures are all equal, namely, one in $S^5$ and one in $S^9$.

Cartan ends his third paper on the subject (Sur quelques familles remarquables d'hypersurfaces, C. R. Congres Math. Liege (1939), 30-41. Also in: Oeuvres Completes, Partie I11, Vol. 2, 1481-1492.) with three questions, one of which asking whether there exist isoparametric hypersurfaces in spheres with $g>3$ such that multiplicities of principal curvatures are unequal. In 1971, Hsiang and Lawson published a paper (Minimal submanifolds of low cohomogeneity, J. Differential Geom. 5 (1971), 1-38.) including a classification of (maximal) subgroups of $SO(n+2)$ whose principal orbits have cohomogeneity $1$ in $S^{n+1}$, and remarked that they precisely coincide with the linear isotropy representations of symmetric spaces of rank two. In 1972, Takagi and Takahashi (On the principle curvatures of homogeneous hypersurfaces in a sphere, Differential Geometry, in Honor of K. Yano, Kinokuniya, Tokyo (1972), 469--481.) remarked that Hsiang-Lawson's result yields a classification of homogeneous isoparametric hypersurfaces in spheres and computed their invariants; in particular, they found examples with $g=4$ and unequal multiplicities.

The relation is that the principal orbits of the linear isotropy representations of symmetric spaces rank two (resp. arbitrary rank) yield beautiful examples of isoparametric hypersurfaces (resp. submanifolds) in spheres.

This relation is relatively easy to explain today. Cartan was the master of both subjects in the late 1930's. Is there anything interesting that can be said about the situation of differential geometry and Lie group theory at that time that prevented him to grasp this connection?

  • 22
    $\begingroup$ This question reminds me of the many times I've seen something done by someone else and thought "Why didn't I think of that?" (because it seemed so obvious once it was pointed out); usually, I couldn't come up with any better answer than that I was thinking along different lines or about different examples. I imagine that even Cartan had this experience from time to time, and maybe he didn't have a better answer either. You can't think of everything, and some things are only 'obvious' in hindsight. $\endgroup$ Feb 9, 2014 at 23:13

1 Answer 1


I think this question is a little subject to opinion, so hopefully it would be considered okay to share mine.

At that time there were a lot of different approaches for how to fit Lie groups into a broader algebraic context. The end of the 1800s saw much debate over things like this (a nice book on this by Crowe: https://www.researchgate.net/publication/244957729_A_History_of_Vector_Analysis). I would say it took at least 40 years to clean things up into a more streamlined system. And some elements of it still merit looking back at the original sources! (I've gotten some interesting results of my own by updating ideas from the early 1900s into modern methodology.)

Sometimes just comparing one person's notation and terminology with another's brings out enlightening relationships that, in hindsight, seem obvious. But carrying out those comparisons is non-trivial and often takes a long time to occur.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.