**Phillip E. Parker
Mathematics Department
Wichita State University
**

In the beginning there was geometry, and it was codified by Euclid
about 300
BC. Some 1500 years later, the contemporary thought was that there
could be only this one, true geometry. This culminated in Kant's argument
in his *Critique of Pure Reason* in 1781 that Euclidean geometry was
an *a priori* synthetic truth. Others appealed to experience or
relied on innate truths, but almost all philosophers and mathematicians
were agreed.

Unfortunately for their certitude, however, Lambert had already proved the existence of a noneuclidean geometry in 1766, but this was not generally known until somewhat later [10] and was not recognized until much later. In 1799, Gauss was already doubting the privileged rôle of Euclidean geometry, and by 1817 his doubt had become assured [8]. Meanwhile, Schweikart had found a noneuclidean geometry by 1816 [16], but did not take the final step of observing that it could not be determined if the universe were Euclidean or not. This remained for Bolyai [3], beginning about 1820, and independently Lobachevsky [11], about 1826.

Thus by 1840 it had become clear to the experts that there were at least
two geometries. In 1854, Riemann gave his inaugural lecture at Göttingen
on the topic of Gauss's choice: the foundations of geometry [15]. He
presented the audience with an infinity of new geometries, those now called
*metric geometries*.

In this lecture, Riemann also gave what turned out to be the notion of
*space* most suitable for geometry. This was an ``*n*-fold extended
quantity," now called a *manifold of dimension* *n*. In dimension 3, a
manifold can be envisaged as various blobs of ordinary space glued
together in a precisely specified way. Clearly, there are a lot of
these spaces.
On each such space, one then specifies a geometry by means of an auxiliary
object called a *metric*.

Some harbored doubts about the ``truth'' of noneuclidean geometries for several years, but by 1872 when Klein gave his inaugural lecture at Erlangen [9] they were mostly assuaged. He enlarged the world of geometries yet again in another major way, declaring that a geometry is the study of those properties which are preserved by a group of transformations, in any space, whether metric or not.

As one might guess, Riemann's and Klein's notions of a geometry do not coincide.
Both are very extensive theories, each including vast arrays of examples of
more or less practical applicability. The part in common has turned out to be
the best place to test our further understanding of geometrical concepts and
notions. The particular subset of the common part where they most closely mesh
consists of those groups which are simultaneously manifolds and in which the
group somehow describes its own geometry *via* a metric. Thus we
combine Riemann's and Klein's geometries on the same space.

Now each group has an operation by which one may compose two elements and
obtain a third. If the order in which two elements are composed does not
affect the outcome, the group is called *commutative*. Thus in a
commutative group, which element is to the right and which is to the
left in a composition does not matter. But the operations of these
geometric groups
are *not* commutative in general, so one must make a choice of
whether one will write down the
left- or right-handed version of the theory. It has become traditional to write
down the left-handed version and leave it to the reader as an exercise to work
out the right-handed version, if desired. Because the geometry is preserved by
or remains invariant under the group operation, one then speaks of
left-invariant geometries on groups which are simultaneously manifolds.
The Norwegian mathematician S. Lie was the first to study such groups
extensively, so they are called *Lie groups*. Thus we arrive at
left-invariant metrics on Lie groups as the tightest combination of the
geometries of Riemann and Klein.

Traditionally, only some of these metric geometries were studied: the
so-called *definite* ones. These are the most natural generalizations
of Euclidean geometry. A summary of the state of knowledge in
1976 may be found in [13]. As early as 1905, however, some applications in
physics required the use of the much larger class of *indefinite*
metric geometries. Unfortunately, very little is yet known about indefinite
left-invariant metrics on Lie groups, and only one particular type has been
studied in any generality [1, 14]. In dimension 3, we have
determined all possible left-invariant metric geometries on Lie groups
[4]. In this dimension there are only two types; but in each higher
dimension, there are many types of indefinite metrics while there is essentially
only one type of definite metric.

One class of Lie groups that is most tractable consists of the *nilpotent*
Lie groups. In a certain technical sense, they are the ones that are the
closest to being commutative. The left-invariant metric geometries on
commutative Lie groups are Euclidean geometry and its nearest indefinite
relatives. Thus the left-invariant metric geometries on nilpotent Lie
groups are those that are closest to being familiar while still exhibiting
distinctive new features. This makes them an ideal place to enhance our
limited understanding of indefinite metric geometric concepts and notions.

Among the nilpotent Lie groups, the very nearest to the
commutative groups are those called *2-step*. In recent years, some of the
most exciting new results in definite metric geometries have been obtained
with 2-step nilpotent Lie groups; *e.g.,* [5]. The definite
left-invariant metric geometry of these was recently studied in some detail
[7], and this is a focus of continuing research; *e.g.,* [12].

We are now studying indefinite left-invariant metric geometries on 2-step nilpotent Lie groups. This is another step in a long-term program to increase significantly our knowledge of the general features of indefinite metric geometries. Other recent parts of this program include [2, 4, 6]. Indefinite left-invariant metric geometries on 2-step nilpotent Lie groups exhibit several new phenomena that do not occur in the definite case. Some of these provide links to other areas of current mathematical research, such as splitting of foliations or decoupling of systems of differential equations, thereby enriching much more than just geometric study.