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?
Best Answer
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.