Let $M$ be a submanifold of a symmetric space $Q$. The normal bundle $NM$ is called *abelian* if $\exp(N\_{p}M)$ is contained in some totally geodesic and flat submanifold of $Q$ for all $p \in M$; see Terng & Thorbergsson, "Submanifold geometry in symmetric spaces", J. Differential Geom. 42 (1995), 665–718.


It is clear that every codimension-one submanifold (i.e., hypersurface) of $Q$ has abelian normal bundle.


I am interested in the case where $Q$ is a Lie group $G$ equipped with a bi-invariant metric.


**Questions**: Are submanifolds with abelian normal bundle (and codimension greater than one) plentiful or rare in $G$? In particular, do two-dimensional examples exist?



Submanifolds of Lie groups with abelian normal bundle
One class of examples in a compact, connected Lie group $G$ of rank $r$ are the conjugacy classes of codimension $r$. Taking an element $a\in G$ whose centralizer is a maximal torus (a generic condition), the conjugacy class of $a$ is a submanifold $C\_a\subset G$ of codimension $r$ whose normal plane at $a$ is the tangent at $a$ to $Z\_a$, the centralizer of $a$ in $G$, which is a flat, totally geodesic submanifold. By the $G$-homogeneity of $C\_a = G/Z\_a\subset G$, the fact that this holds at $a$ implies that it holds at all point of $C\_a$.


This gives an $r$-parameter family of mutually noncogruent examples.


There are other examples: When $G=\mathrm{SU}(n)$, and $a = \mathrm{diag}(\lambda\_1,\lambda\_2,\ldots,\lambda\_n)$ is a diagonal element for which the $\lambda\_i^2$ are all distinct, the submanifold
$$
M\_a = \{\ g\_1 a g\_2\ |\ g\_1,g\_2\in\mathrm{SO}(n)\ \}\subset\mathrm{SU}(n)
$$
is homogeneous under the isometry group of $\mathrm{SU}(n)$ (endowed with its biïnvariant metric), and its tangent plane at $a$ is orthogonal to the diagonal maximal torus $T\subset\mathrm{SU}(n)$, so, by homogeneity, its tangent plane at any point is orthogonal to a flat, totally geodesic submanifold of $\mathrm{SU}(n)$. Hence it has an abelian normal bundle.


This gives another $r=n{-}1$ parameter family of mutually noncongruent examples of codimension $r$, distinct from the conjugacy classes.


Using the methods of exterior differential systems, one can show that, when $n=3$, these two families account for all of the codimension $2$ submanifolds of $\mathrm{SU}(3)$ with abelian normal bundle, in the sense that any connected submanifold $M^6\subset\mathrm{SU}(3)$ with abelian normal bundle is, up to ambient isometry, an open subset of one of the examples listed above. The argument that I have written out is a calculation using exterior differential systems and the moving frame, but, when I have time, I can sketch the proof, if there is interest.


**Addendum 1:** I had a flight with some time to look at the other two compact simple rank 2 groups. It turns out that all of the connected codimension two submanifolds with abelian normal bundle in $\mathrm{SO}(5)$ and $\mathrm{G}\_2$ are (open subsets of) homogeneous compact ones as well. In each case, there is one additional $2$-parameter family of examples beyond the principal conjugacy classes.


**Addendum 2:** I also checked the rank $r=3$ case $G = \mathrm{SU}(4)$, and found that every connected codimension $3$ submanifold of $G$ with abelian normal bundle is an open subset of one of the two types of homogeneous examples listed above. Since any codimension $2$ submanifold of $\mathrm{SU}(4)$ that is foliated by codimension $3$ submanifolds of $\mathrm{SU}(4)$ with abelian normal bundle will have abelian normal bundle, it follows that there are many non-homogeneous codimension $2$ submanifolds of $\mathrm{SU}(4)$ with abelian normal bundle.


In fact, it now seems likely that, for any compact simple group $G$ of rank $r>1$, there are two $r$-parameter families of homogeneous codimension $r$ submanifolds with abelian normal bundle and every connected codimension $r$ submanifold with abelian normal bundle is, up to ambient isometry, an open subset of a homogeneous one. The two families are as follows: The first family is the family of principal conjugacy classes in $G$, and the second family is the set of principal orbits of $K\times K$ acting by left and right multiplication in $G$ where $G/K$ is a symmetric space of rank $r$. For example, when $G=\mathrm{SU}(n)$ $(n\ge3)$, $K=\mathrm{SO}(n)$; when $G = \mathrm{SO}(n)$ $(n\ge 5)$, $K = \mathrm{SO}(p)\times\mathrm{SO}(n{-}p)$ where $p = \bigl[\tfrac12 n\bigr]$; when $G=\mathrm{Sp}(n)$ $(n\ge3)$, $K=\mathrm{U}(n)$; when $G=\mathrm{G}\_2$, $K=\mathrm{SO}(4)$; when $G=\mathrm{F}\_4$, $K=\mathrm{Sp}(3)\mathrm{SU}(2)$; when $G=\mathrm{E}\_6$, $K=\mathrm{Sp}(4)$; when $G=\mathrm{E}\_7$, $K=\mathrm{SU}(8)$, and when $G=\mathrm{E}\_8$, $K=\mathrm{SO}'(16)$.


