Phillip E. Parker
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  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 . Meanwhile, Schweikart had found a noneuclidean geometry by 1816 , 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 , beginning about 1820, and independently Lobachevsky , 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 . 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  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 . 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 . 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., . The definite left-invariant metric geometry of these was recently studied in some detail , and this is a focus of continuing research; e.g., .
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.
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