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\begin{center}
{\Large {GOLDEN OLDIE 18}} \\
\bigskip
\bigskip
{\Large{On the three-dimensional spaces which admit a continuous group of
motions}}
\bigskip
\bigskip
\begin{center}
{\bf Luigi Bianchi}
{\it Memorie di Matematica e di Fisica
della Societa Italiana delle Scienze, Serie Terza}, Tomo {\bf XI},
pp.~267--352
(1898).
\end{center}
\bigskip
\bigskip
EDITOR'S NOTE \\
\end{center}
This article methodically studies (locally) the symmetry and isometry
classes
of all 3-dimensional Riemannian manifolds. For each of the possible orbit
dimensions 1 and 2 (intransitive actions) and 3 (transitive actions) and for
each possible symmetry class of group actions, explicit canonical coordinate
expressions are derived for the full Killing vector Lie algebra and the
metric
by solving the Killing equations. A representative line element is then
given
parametrizing the isometry classes of a given symmetry type modulo constant
conformal transformations, and specializations which admit higher symmetry
are
studied. For the case of simply transitive 3-dimensional isometry groups,
this
classification of metrics by symmetry class coincides with the
classification
into nine isomorphism classes of the isometry groups themselves (Bianchi
types
I -- IX), now known together as the Bianchi classification.
This article followed soon after Lie's classification over the complex
numbers
of all Lie algebras up to dimension 6 and Killing's discovery of his famous
Killing equations at the end of the nineteenth century. All of Bianchi's
work
was well known by the mathematician Luther P. Eisenhart (1876--1965), a
professor, chair, dean and important educator in the Princeton University
Mathematics Department from 1900 to 1945 \cite{gillispie}, who served as a
principal source of English language discussion of much of the early work in
Riemannian geometry and Lie group theory through his two books {\it
Riemannian
Geometry\/} (1925) \cite{eisenhart1} and {\it Continuous Groups of
Transformations\/} (1933) \cite{eisenhart2}, which contain numerous
references
to Bianchi's work. As a differential geometer, Eisenhart occasionally helped
Einstein and certainly contributed to the enthusiasm for relativity at
Princeton.
The results of this Bianchi article were extended and brought to the attention
of the relativity community in 1951 by Abraham Taub just after their first use
in two special applications by G\"odel. Taub got his Ph.D. in Mathematics at
Princeton University in 1935 under H.P. Robertson, during the time (1933--1939)
in which the Institute for Advanced Study was founded but initially housed in
the Princeton University mathematics building where Taub had learned his
differential geometry from Eisenhart and worked with both Oswald Veblen and
John von Neumann, two of the three mathematicians stolen from the university as
the original members of the Institute with Einstein when it opened in 1933
\cite{regis}. Later G\"odel, Einstein's best friend and a fellow member of the
Institute, was led by philosophical questions about time \cite{goedel,goedelw}
to consider studying rotating universe models in the late 1940s, leading to the
first application of Bianchi's homogeneous spaces in general relativity (types
III, VIII \cite{iiiviii}), shaking up the physics community with the strange
new properties of his stationary rotating solution (1949) \cite{godel1},
followed by a summary of results he published in 1950 without proof about
rotating and expanding cosmological solutions (type IX) \cite{godel2}. Taub was
Veblen's assistant in 1935--36 and a visiting member of the Institute in
1947--48 \cite{tipler} and soon after announced his own work at the same
conference at which results of G\"odel's second investigation were presented,
shortly later in 1951 publishing explicit formulas for the spatially
homogeneous spacetime metrics corresponding to all of Bianchi's nine symmetry
types and the vacuum Einstein equations for these cosmological models in a
discussion (like G\"odel) motivated by the desire to find solutions violating
Mach's principle \cite{taub}.
These Bianchi models, as they later came to be called, were revived in the late
1950's by Heckmann and Sch\"ucking (later summarized in their chapter
\cite{heckschuck} in {\it Gravitation, an Introduction to Current Research\/}
edited by Louis Witten, father of Edward). Istv\'an Ozsv\'ath continued this
work in the next decade \cite{ozssch, ozs}, during which time David Farnsworth
and Roy Kerr (1966) introduced the modern Lie group description of homogeneous
spaces in relativity \cite{farkerr}, while C.G. Behr (1968) introduced the
modern Lie algebra version of the Bianchi (Lie) classification of 3-dimensional
Lie algebras using the irreducible parts of the structure constant tensor under
linear transformations \cite{ewb}. Meanwhile Ronald Kantowski (1966) explored
for the first time the spatially homogeneous (Kantowski-Sachs) models with no
simply transitive subgroup \cite{kantowski}, a spatial geometry thoroughly
studied by Bianchi (\S9) but for some reason omitted in his final summary,
while Ellis (1964, 1967) pioneered the application of modern tetrad methods to
cosmological models in his study of locally rotationally symmetric dust,
involving the whole class of Bianchi symmetry types admitting multiply
transitive groups.
At the close of the 1960's an ongoing investigation into the nature of the
initial singularity of the universe by the Russian school of Lifshitz and
Khalatnikov, later joined by Belinsky, independently led to the general Bianchi
models in describing how the spacetime metric behaves along timelike curves
approaching a ``generic" spacelike singularity in some limiting approximation
that was then controversial. At about the same time a study of the chaotic
behavior they discovered was begun by Misner, who used Hamiltonian methods to
explore the Bianchi type I and Bianchi type IX (Mixmaster) universe dynamics in
the USA. While Ellis and MacCallum \cite{ellmac} approached the Bianchi models
from an orthonormal frame point of view in England, Misner's Hamiltonian
studies were continued by his student Michael P. Ryan for the general Bianchi
model case, later summarized in 1975 in the only book devoted specifically to
Bianchi cosmology \cite{ryanshepley}, in whose bibliography references to the
above-mentioned but uncited work may be found by year of publication.
Bogoyavlensky and Novikov pioneered the application of the
qualitative theory of differential equations to the dynamics of general Bianchi
models (1973); more recent work in this direction is described in the book {\it
Dynamical Systems in Cosmology\/} \cite{dsc}, where references to their work
may be found.
The Bianchi models are spatially homogeneous spacetimes, the spatial sections
of which are homogeneous Riemannian 3-manifolds of a fixed Bianchi type, and
usually they are interpreted as cosmological models. While generally spatially
anisotropic, they contain the spatially homogeneous and isotropic
Friedmann-Robertson-Walker models as special cases for certain symmetry types,
and enable Einstein's equations or similar gravitational field equations to be
reduced from partial to ordinary differential equations, which are much easier
to study. Besides providing more generalized models of certain aspects of the
early universe, they have also been invaluable in helping to understand
features of general relativity itself by providing an arena where certain
questions can be more easily investigated. The most recent and sophistocated new
work on spatially homogeneous cosmologies and their spatial geometries involves
the Teichm\"uller space analysis of the dynamical degrees of freedom and
Hamiltonian structure for spatially compactifiable models\cite{kth}.
In 1972 (during the ``golden age of relativity" at Princeton) when John
Wheeler
was bringing in proofs of his new book {\it Gravitation\/} with coauthors
Charles Misner and Kip Thorne \cite{mtw} to my sophomore Modern Physics
class
at Princeton University, junior Jim Isenberg was recruiting students to fill
the quota for a student initiated seminar on Differential Geometry for
General
Relativity to be offered by Wheeler's collaborator Remo Ruffini. Following
Wheeler's teaching style, Ruffini wanted to get the students more involved
by
doing special projects, and one suggestion was for a student to help him
translate into English the original papers of Bianchi on homogeneous
3-spaces.
Somehow I volunteered, but it immediately became clear that this was very
inefficient so I boned up on some elementary Italian based on 3 prior
semesters of college Spanish and tackled the job during the summer while
working during the day as a carpenter with my dad.
This was followed by a junior paper on Bianchi cosmology and later a senior
thesis begun in 1973 when Ruffini (my advisor) was excited by his
investigation
of the orbits of particles in rotating black hole spacetimes with another
undergraduate from his seminar (Mark Johnston, whose graphics led to the
famous
Marcel Grossmann Meeting logo). Curious about rotating cosmologies, Ruffini
wondered about talking to G\"odel himself about the problem. Looking him up
in
the phone book (still naive times for celebrities), Ruffini found him,
called
him up and arranged for me to meet him at his office at the Institute, where
he
informed me about recent work by Michael Ryan that I had not been aware of,
initiating my own work in the dynamics of Bianchi cosmology. G\"odel, though
his only published work in relativity was over 20 years old at the time, had
still been following current developments related to it.
Ruffini later channeled me toward grad school at the University of
California
at Berkeley to work with Abe Taub just before his retirement in 1978.
However,
the Bianchi translation, although it had been typed up by a Jadwin Hall
secretary in 1973, never found a wider audience, and sat for 25 years until
Andrzej Krasi\'nski asked me if I might translate the long article for this
series, not knowing that it had essentially already been done (but which
needed
conversion to a compuscript and a polishing of my translation with my
Italian
improved by 20 years of regular visits to Rome).
Unfortunately by this time
(1999), Taub was in failing health and then passed away and could no longer
be
consulted to unravel more of the interesting history of the personalities at
Princeton tied together by Bianchi's work. However, this question led to my
volunteer project to put {\it The Princeton Mathematics Community in the
1930s:
An Oral History Project\/} \cite{gill2} on the Princeton University Library
web
together with background materials that should be of interest to those
curious
about the community that welcomed Einstein with the founding of the
Institute
for Advanced Study in Princeton.
Finally this project could not have been completed (2001) without \LaTeX, which
allowed me to typeset a long formula-dense document, nor without the
encouragement and invaluable editorial assistance of Andrzej Krasi\'nski, who
went beyond the call of duty in many rounds of proofreading comparing my
document to the original Italian manuscript to ensure as accurate a
reproduction as possible in every detail.
\subsection*{Commentary on Bianchi's Article in Translation}
\paragraph{Terminology}
Bianchi uses the term ``group" to mean ``transformation group" or a group
action on a manifold, expressed in terms of local coordinates on the group
manifold and the manifold on which it acts, and he specifies such groups $G_r$
(dimension $r$) by giving a basis of the Lie algebra of generating vector
fields (called ``infinitesimal transformations"), using the notation $G_r
\equiv (X_1 f, \ldots, X_r f)$, where $X f = \sum_{i}^{1..n} \xi_i\partial
f/\partial x_i$ denotes the action of a vector field on an arbitrary function
$f$ by differentiation and his square bracket delimiters have been replaced by
parentheses to lessen confusion with the modern commutator notation. Coordinate
(and index) labels are subscripted in his notation: $x_i$. The modern
Christoffel symbols of the first and second kind $[ij,k]$ and $k\brace{ij}$
have replaced the original symbols ${ij}\brack k$ and ${ij}\brace k$ in use at
the time (apparently introduced by Eisenhart to conform with the Einstein index
convention \cite{eisenhart1}). Bianchi's commutator (Lie bracket) notation
$(X_1 X_2) = X_1 f$ which uses parentheses but no comma, with no arbitrary
function to the right of the commutator (although both comma and function
appear in his later work \cite{bianchibook}), has been modernized to the square
bracket convention $[X_1, X_2]f = X_1 f$. Two equivalent group actions (in the
coordinate representation: related by invertible joint coordinate
transformations on the group manifold and manifold on which it acts), are
called ``similar" by Bianchi, while two metrics are called ``similar" if they
are locally isometric modulo a constant conformal factor. (Bianchi uses the
term ``applicable", which has been updated to ``isometric.") The
transformations of a group acting as isometries of a metric are called
``motions." The version of this article published in his collected works has
more complete footnotes (consecutively numbered, first name initials added)
which have been used here, and a correction in proof (rewording of the
beginning of the next to last sentence of \S21) was incorporated into the text
as done there, together with a correction to the sign of equation (62b) which
allowed the deletion of several lines at the end of \S19, and a few other minor
corrections. Multiple equations grouped together by an expanded left brace
delimiter in the original have been distinguished here by a letter following
the equation number or the brace has simply been omitted when unnecessary, and
some displayed equations have been incorporated into the text. (As a
consequence of this modification of equation numbers, the references to those
numbers in the text were also modified but not marked by footnotes.) Finally a
number of index typo's from the original articles have been corrected in
translation.
The group generated by the commutators of the generating vector fields is
called the derived group. The numerical scheme characterizing the Bianchi
classification of simply transitive 3-dimen\-sion\-al group actions is a simple
one based on the sequence of successive derived groups starting from the
original one, as described in \S198 (I sette tipi di composizioni dei $G_3$
integrabili, The seven types of compositions of integrable $G_3$'s) and \S199
(I due tipi dei $G_3$ nonintegrabili, The two types of nonintegrable $G_3$'s)
of his book on continuous groups \cite{bianchibook}. This is Bianchi's version
of Lie's classification of equivalence classes of 3-dimensional Lie algebras
over
the complex numbers,
refined for equivalence over the real
numbers. (In a similar way Lie essentially classified all Lie algebras up to
dimension 6, with the 4-dimensional case done explicitly in all detail.)
Luther
Eisenhart's book \cite{eisenhart2} is an excellent source of information for
the terminology and Lie group details of this early generation of geometers.
\paragraph{Preliminaries}
The article begins with the Killing equations for an
$n$-dimen\-sion\-al Riemannian manifold (\S1) and briefly treats the
one Killing vector case (\S2).
(Bianchi uses the notation $X$ for $\pounds_X$.)
Then the classification of 2-dimen\-sion\-al
Riemannian manifolds with simply transitive isometry groups ($G_2$)
is reviewed (\S3), with only two discrete transformation group types:
Abelian and non-Abelian, leading to the negative and zero curvature
cases, both of which have 3-dimen\-sion\-al complete isometry groups.
Since the derived group of a $G_2$ generated by the commutators of the
generating vector fields must be 0 or 1-dimensional in two dimensions,
choosing $X_1$ to span its Lie
algebra gives the canonical form of the non-Abelian case commutation
relations:
$[X_1,X_2] = \epsilon X_1$, with $\epsilon=1$. The Killing vectors
$(X_1,X_2)$ =
($e^{-\epsilon x_2}\partial/\partial x_1$,$\partial/\partial x_2$)
and the general forms of the metric and the invariant 1-forms are derived
but not explicitly stated:
$$
ds_{(2)}^2 = \alpha \,(dx_1 + \epsilon x_1 dx_2)^2
+ 2\beta \,(dx_1 + \epsilon x_1 dx_2)\, dx_2
+\gamma \, dx_2^2\ ,
$$
with $\epsilon=1$ describing the non-Abelian case and
$\epsilon=0$ the Abelian case.
This is then used in the case of 3-dimensional Riemannian
manifolds with 2-dimensional intransitive isometry groups
($G_2 \equiv (X_1 f,X_2 f)$) acting simply transitively
on 2-dimensional orbits (\S4). The orbits are a family of
geodetically parallel surfaces
taken as $x_1$ coordinate surfaces in an adapted Gaussian
normal coordinate system
with orthogonal geodesics along the coordinate lines of $x_1$,
while $x_2,x_3$ are adapted to the generators as above,
leading to the general form
$$
ds^2 = dx_1^2 +
\alpha \, (dx_2 - \epsilon x_2 dx_3)^2
+ 2\beta \, (dx_2 - \epsilon x_2 dx_3)dx_3
+\gamma \, dx_3^2\ ,
$$
where the three independent components
$\alpha$, $\beta$, $\gamma$ of the 2-metric
in the invariant form basis are functions only of $x_1$.
Then the complete isometry groups possible for such
intransitive actions (2-dimensional
orbits) are described (\S\S5--11), which can only be
at most 3-dimensional, forcing the
surfaces to have constant curvature. This is the case
of intransitive groups acting multiply
transitively on 2-dimensional orbits. However, the
additional isometries can lead
to a transitive action, which is the case for the
4-dimensional isometry groups of the
positive (\S9) and negative (\S11) curvature
Kantowski-Sachs geometry, or the 6-dimensional
isometry groups of constant positive (\S8), zero (\S8),
or negative (\S\S8,10) curvature spaces.
\paragraph{Homogeneous 3-Manifolds}
These preliminary considerations are then used in the case
of 3-dimensional simply transitive
isometry groups. Such simply transitive actions are first
introduced for any dimension,
with a discussion of the coordinate representation of the
Killing equations and their
integrability conditions, the latter being satisfied
identically by virtue of the
generating Lie algebra Lie bracket relations (\S12).
Then Lie's classification of
equivalence classes of 3-dimensional Lie algebras over
the complex numbers is refined
to the real case by adding several types, giving Bianchi's
canonical form for the
generating Lie algebra commutation relations for each type
designated by consecutive
Roman numerals I through IX, now known as the Bianchi
classification (\S13).
Types I through VIII all have a 2-dimensional subgroup
$G_2\equiv(X_1 f,X_2 f)$
acting simply transitively on 2-dimensional orbits, so
the metric can be reduced to
the standard form given above for intransitive actions
on surfaces,
with types I through VII belonging to the Abelian subgroup
case and
type VIII to the non-Abelian subgroup case.
Equations (E) of \S14 are Killing's equations for this metric
in the Abelian subgroup case,
then applied in (F) to the third generating vector field $X_3$
whose Lie brackets with
$X_1$, $X_2$ are given by equations (41). Specializing this
pair of sets of equations
to each symmetry type then leads to the complete symmetry group,
including the coordinate
representation of $X_3$ and additional independent Killing
vector fields, and to
explicit values for the three metric coefficient functions
of $x_1$,
from which one may easily read off the invariant 1-forms
in terms of which the metric is expressed, though not done
explicitly by Bianchi.
The coordinate expressions for the metric and Killing vector
fields are then specialized
(by rescaling the surface coordinates $x_2,x_3$, by affine
transformations of the surface parametrizing coordinate $x_3$,
and by constant conformal transformations) to a simple canonical form
with the minimum number of free parameters, which then
parametrize the conformal
equivalence classes of homogeneous 3-geometries (locally).
The symmetry type I case of flat 3-space in orthonormal
Cartesian coordinates with its
translation symmetries is trivially obtained from these
equations, with a 6-dimensional
complete isometry group.
For the symmetry type II (\S16), all metrics are conformally
equivalent and have a
4-dimensional complete isometry group whose finite equations
are given, corresponding to right multiplication of a unit
upper triangular $3\times3$ matrix
$X(x_1,x_2,x_3) = I_3 + x_3 e^2{}_1 - x_2 e^3{}_1 + x_1 e^3{}_2$
by $A(-a_1,-a_2,-a_3)^{-1}$.
For the symmetry type III (\S17), a 1-parameter family of conformal
equivalence
classes is found, with the parameter $n$ measuring the nonorthogonality of
the
surface coordinates $x_2,x_3$, and a 4-parameter complete group of
isometries
whose derived group $(X_1 f,X_3 f, X_4 f)$ is of type VIII, which acts
transitively when $n\neq0$ and intransitively when $n=0$ (therefore
appearing
in the discussion of intransitive actions). The additional linearly
independent
Killing vector field $X_4 f$ depends on $n$. The proof that $n$ parametrizes
the conformal 3-geometry involves showing that two metrics with the same
canonical form in two coordinate systems but with different values $n$ and
$m$
of the parameter cannot be related by a coordinate transformation. The two
4-dimensional isometry groups must be equivalent by a theorem of Lie, but
the
accompanying canonical generating vector fields need only be transformed
into
each other by the coordinate transformation modulo a Lie algebra
automorphism.
The 4-parameter group of Lie algebra automorphisms is easily found, and with
some more work, a 3-parameter group of coordinate transformations which
transform the generators into each other, giving the equivalence
transformation
between the two isometry groups (\S18). The partial derivatives of one set
of
coordinates with respect to the other can be read off from the
transformation
of the generating vector fields, and used to evaluate the transformation of
the
one metric into the other. Requiring the two metrics to be related by the
same
coordinate transformation modulo a constant conformal factor then forces the
two parameters $n$ and $m$ to be the same modulo an irrelevant sign (\S19).
As
noted above, the form (49) of the type III metric, changed in signature,
slightly rescaled and with a special value of $n$, was used by G\"odel for
the
timelike homogeneous sections of his stationary spacetime homogeneous
solution.
For the symmetry type IV (\S20), similarly a 1-parameter
family of conformal equivalence
classes is found with no additional Killing vector fields.
The 4-parameter group of
Lie algebra automorphisms is easily found, and then a
5-parameter family of coordinate
transformations which induce them, and the essential
nature of the parameter is again
shown by transforming the metric (\S21).
The symmetry type V immediately leads to an orthogonal
coordinate representation of
the constant negative curvature geometry with a 6-dimensional
complete isometry group
(\S22) whose generators are derived as an example in \S38.
The symmetry type VI (\S\S23,24) is entirely analogous to the type IV case.
Bianchi does not distinguish the modern class A and class B types VI$_0$ and
VI$_{h\neq0,-1}$, where the subscript $h$ is the Behr parameter described
below, differing from Bianchi's parameter $h\neq0,1$. Bianchi's $h=-1$, $h=0$
and $h=1$ limits of type VI give types VI$_0$, V and III respectively.
The symmetry type VII is split into types VII$_1$ and VII$_2$, corresponding to
VII$_0$ and VII$_{h\neq0}$ in the modern Behr notation but with a different
parameter $h$. The metric coefficients and $X_3$ are found (\S\S25,26), again
with a 1-parameter family of conformal equivalence classes and with no
additional Killing vector fields (except for the special case of type VII$_0$
corresponding to flat 3-space), and then a 4-parameter Lie algebra automorphism
group is found and used to show the essential nature of the parameter (\S27).
The symmetry type VIII then switches to the non-Abelian $G_2\equiv(X_2 f,X_3
f)$ subgroup case (\S28), where two cases arise. The simpler case with an
additional Killing vector field is equivalent to the type III case $n\neq0$,
but the more general case in which no additional Killing vectors exist, the
Gaussian normal coordinates lead to elliptic functions in the integration of
the Killing equations for the metric coefficients and the third generator
$X_1$
(\S\S29,30), where Bianchi's original notation for the elliptical functions
lacked parentheses around their arguments. By choosing a more general
coordinate $x_1$ whose coordinate lines are no longer orthogonal to the
2-surface orbits of the $G_2$ and which does not measure arclength along
them,
but for which $X_1$ has a relatively simple form ($A=1$, $B=C=0$ in equation
(95) for $X_1$), expressions are found for the metric coefficients which are
at
most quadratic in the two nontrivial coordinates separately (\S31).
Finally the symmetry type IX requires a similarly different
approach. Canonical
generating vector fields long known from Euler angle
parametrizations of the
rotation group are chosen and the Killing equations integrated
to yield a 6-parameter
family of metric coefficients from which one could read off
the invariant 1-forms (\S32).
No discussion of conformal equivalence classes is given.
Relying again on Lie, the case in which an additional independent
Killing vector field $X_4$ exists is treated, leading to a
1-parameter conformal equivalence class of metrics which includes
the special case of constant positive curvature (and a
6-dimensional complete isometry
group) for a particular value of the parameter (\S33),
whose essential nature is shown
in \S35 after showing that no additional Killing vectors
exist other than $X_4$ (\S34).
That no other possibilities have been overlooked is shown in \S36, relying
on
the the fact that no 5-dimensional isometry groups can exist as shown in
\S37.
The final section summarizes the canonical form of the metric, Killing
vector
fields, and their Lie brackets for most of these possibilities, although the
4-dimensional isometry group with no simply transitive 3-dimensional
subgroup
case of \S11 is curiously omitted, perhaps leading to the nearly two decade delay in its
application to spatially homogeneous cosmological models, first done by
Kantowski and Sachs \cite{kantowski}.
\paragraph{Obtaining the Same Results Painlessly from a
Modern Perspective}
At least in the general relativity literature, Farnsworth and Kerr
\cite{farkerr} first published the modern description of a simply transitive
symmetry group action as the natural left or right action of any Lie group
on
itself, moving the jargon away from the old fashioned simply transitive
transformation group accompanied by an isomorphic reciprocal group to left
and
right translation on the group manifold. Choosing a left action for the
symmetry action, the left invariant vector fields (Lie algebra of the Lie
group) are the homogeneous (``invariant") vector fields, the left invariant
(positive-definite) metric tensors on the group manifold are the homogeneous
Riemannian metrics, with the corresponding Killing Lie algebra for this
``homogeneity" action equal to the Lie algebra of right invariant vector
fields. For spatially homogeneous spacetimes (``Bianchi cosmologies"), the
induced metrics of the spatial hypersurfaces of homogeneity are isometric to
left invariant Riemannian metrics on a fixed 3-dimensional spacetime
isometry
group.
Behr \cite{ewb} was the first to then publish a simpler scheme for
classifying
the equivalence classes of 3-dimensional Lie algebras using the irreducible
parts of the structure constant tensor under linear transformations rather
than
the more complicated derived group approach of Lie and Bianchi, exploiting
the
special properties of the duality operation in 3 dimensions: taking the
natural
dual of the covariant antisymmetric indices of the structure constant tensor
$C^a{}_{bc}$ leads to a 2-covariant tensor density on the Lie algebra
$C^{ab}=\frac12 C^a{}_{cd} \epsilon^{bcd}$
which can be decomposed into its
symmetric $n^{ab}=C^{(ab)}$ and antisymmetric parts $C^{[ab]}=\epsilon^{abc}
a_c$, and the dual of its antisymmetric part leads to a covector
$a_c=\frac12
\epsilon_{cab} C^{ab}$ which the Jacobi identity shows must have zero
contraction with the symmetric 2-tensor $n^{ab} a_b=0$. When nonzero, this
covector's self tensor product must then be proportional to the matrix of
cofactors of the 2-covariant symmetric tensor with a scalar constant of
proportionality $a_a a_b=\frac12 h \epsilon_{acd}\epsilon_{bfg} n^{cf}
n^{dg}$.
This notation was introduced by Ellis and MacCallum \cite{ellmac}, who also
coined the terms class A for the case $a_b=0$ and class B for the case
$a_b\neq0$ (corresponding to unimodular and nonunimodular Lie algebras
\cite{milnor}). Diagonalizing the symmetric tensor density $n^{ab}$ aligns
$a_b$ with one of the basis vectors in general, leading to a standard
``diagonal form" for the 4 nonzero components of the structure constant
tensor
(of which at most 3 can be simultaneously nonzero), from which the
equivalence
class representative structure constants are obtained by quotienting out by
the
scale transformations of the Lie algebra basis vectors.
Jantzen \cite{jantzen} realized that explicit expressions for all the
invariant
vector fields and 1-forms, and hence for the homogeneity Killing vector
fields
and the general form of the metric, could be easily obtained from the
generic
expressions for the linear adjoint matrix group associated with a diagonal
form
basis of the Lie algebra, which generically has the same Lie algebra
structure
as the original Lie algebra in 3 dimensions, with limiting cases following
by
analytic continuation of the formulas valid in the general case. Similarly
by
considering the orbits of the easily constructed automorphism matrix groups
on
the space of inner products on the Lie algebra, one can algebraically
determine
the isometry classes of individual Bianchi symmetry types. This almost
eliminates the need for solving any partial differential equations, the
element
responsible for the length of Bianchi's classification paper.
\begin{thebibliography}{00}
\bibitem{gillispie}
Charles C. Gillispie, editor,
{\it Dictionary of Scientific Biography\/},
Scribner's Sons, New York, 1970.
\bibitem{eisenhart1}
Luther P. Eisenhart,
{\it Riemannian Geometry\/},
Princeton University Press, 1925.
\bibitem{eisenhart2}
Luther P. Eisenhart,
{\it Continuous Groups of Transformations\/},
Princeton University Press, 1933.
\bibitem{regis}
Ed Regis,
{\it Who Got Einstein's Office?\/},
Addison-Wesley, New York, 1987.
\bibitem{goedel}
John W. Dawson, Jr, {\it Logical Dilemmas: The Life and Work of Kurt
G\"odel\/},
A.K. Peters, Wellesley, Massachussetts, 1997, Chapter 9.
\bibitem{goedelw}
{\it Kurt G\"odel: Collected Works\/}, Vols. I (1986), II (1990), edited by
Solomon Feferman, Vol. 3 (1995), edited by Solomon Feferman, John W. Dawson,
Jr., Warren Goldfard, Charles Parsons and Robert M. Solovay, The Clarendon
Press, Oxford University Press, New York; see especially the unpublished
manuscript of his famous Institute for Advanced Study lecture of May 7, 1949 in
Vol.~3 and his preceding discussion of the relationship between relativity and
Kantian philosophy which motivated his cosmology work.
\bibitem{iiiviii}
Istv\'an Ozsv\'ath, ``Dust-Filled Universes of Class II and Class III," {\it
J.\ Math.\ Phys.\/} {\bf11}, 2871 (1970); Robert T. Jantzen, ``Generalized
Quaternions and Spacetime Symmetries," {\it J.\ Math.\ Phys.\/} {\bf 23},
1741
(1982); G\"odel's Lorentzian 3-metric obtained setting $dx_3=0$ (a trivial
translation symmetry coordinate) coincides with Bianchi's equation (49) with
$(n,dx_3,dx_2,dx_1^2)\rightarrow (\sqrt{2},dx_0,dx_2/\sqrt{2},-dx_1^2)$, but
this locally rotationally symmetric type III geometry coincides with a
locally
rotationally symmetric type VIII geometry slightly deformed from the
bi-invariant Cartan-Killing metric on the group, which G\"odel exploits
using a
quaternion-like representation, later discussed by Ozsv\'ath.
\bibitem{godel1}
Kurt G\"odel,
``An Example of a New Type of Cosmological Solutions of Einstein's Field
Equations of Gravitation,"
{\it Reviews of Modern Physics\/} {\bf 21}, 447 (1949); {\it Gen.\ Rel. \
Grav.\/} {\bf 32}, 1409-1417 (2000).
\bibitem{godel2}
Kurt G\"odel,
``Rotating Universes in General Relativity Theory,"
{\it Proceedings of the
International Congress of Mathematicians\/},
Cambridge, Mass. 1950, Vol. 1, 175,
Amer.\ Math.\ Soc., R.I., 1952;
{\it Gen.\ Rel. \ Grav.\/} {\bf 32}, 1419-1427 (2000).
\bibitem{tipler}
{\it Essays in General Relativity: a Festschrift for Abraham Taub\/},
edited by Frank J. Tipler, Academic Press, New York, 1980.
\bibitem{taub}
Abraham Taub,
``Empty Spacetimes Admitting a Three-Parameter Group of
Motions,"
{\it Proceedings of the
International Congress of Mathematicians (Cambridge, Mass., 1950),
p.655\/}; {\it Annals of Mathematics\/} {\bf 53}, 472 (1951); to appear in {\it
Gen.\ Rel. \ Grav.\/} {\bf 33}, (2001).
\bibitem{heckschuck}
O. Heckmann and E. Sch\"ucking,
``Relativistic Cosmology,"
in {\it Gravitation, an Introduction to
Current Research\/}, edited by Louis Witten, Wiley, New York, 1962.
\bibitem{ozssch}
I. Ozsv\'ath and E. Schucking,
``Finite Rotating Universes,"
{\it Nature\/} {\bf 193}, 1168 (1962).
\bibitem{ozs}
I. Ozsv\'ath, ``Spatially Homogeneous
World Models," {\it J.\ Math.\ Phys.\/} {\bf11}, 2860 (1970); ``Spatially
Homogeneous Rotating World Models," {\it J.\ Math.\ Phys.\/} {\bf12}, 1078
(1971).
\bibitem{farkerr}
D.L. Farnsworth and R.P. Kerr,
``Homogeneous, Dust-Filled Cosmological Solutions,"
{\it J.\ Math.\ Phys.\/} {\bf 7}, 125 (1966).
\bibitem{ewb}
F.B. Estabrook, H.D. Wahlquist, and C.G. Behr,
``Dyadic Analysis of Spatially Homogeneous World Models,"
{\it J.\ Math.\ Phys.\/} {\bf 9}, 497 (1968).
\bibitem{kantowski}
Ronald Kantowski, ``Some Relativistic Cosmological Models" (Ph.D. thesis 1966),
{\it Gen.\ Rel. \ Grav.\/} {\bf 30}, 1663 (1998); R. Kantowski and R.K. Sachs,
``Some Spatially Homogeneous Anisotropic Relativistic Cosmological Models,"
{\it J.\ Math.\ Phys.\/} {\bf 7}, 443 (1966).
\bibitem{ellmac}
G.F.R. Ellis and M.A.H. MacCallum, ``A Class of Homogeneous Cosmological
Models," {\it Commun.\ Math.\ Phys.\/} {\bf 12}, 108 (1969).
\bibitem{ryanshepley}
Michael P. Ryan, Jr. and Lawrence Shepley,
{\it Homogeneous Relativistic Cosmologies\/},
Princeton University Press, Princeton, 1975.
\bibitem{dsc}
{\it Dynamical Systems in Cosmology\/}, edited by J. Wainwright and G.F.R.
Ellis, Cambridge University Press, Cambridge, 1997.
\bibitem{kth}
T. Koike, M. Tanimoto and A. Hosoya, ``Compact Homogeneous Universes," {\it J.\
Math.\ Phys.\/} {\bf 35}, 4855 (1994);
M. Tanimoto, T. Koike and A. Hosoya,
``Dynamics of Compact Homogeneous Universes," {\it J.\ Math.\ Phys.\/} {\bf
38}, 350 (1997); ``Hamiltonian Structures for Compact Homogeneous Universes,"
{\it J.\ Math.\ Phys.\/} {\bf 38}, 6560 (1997).
\bibitem{mtw}
Charles W. Misner, Kip S. Thorne and John A. Wheeler,
{\it Gravitation\/}, Freeman, San Francisco, 1973.
\bibitem{gill2}
{\it The Princeton Mathematics Community in the 1930s: An Oral History
Project\/}, Charles C. Gillispie, Albert W. Tucker, William Aspray, and
Frederik Nebeker, Princeton University, 1985; now on-line at {\tt
http://www.princeton.edu/mudd/math}.
\bibitem{bianchibook}
Luigi Bianchi, {\it Lezioni sulla teoria dei gruppi continui finiti di trasformazioni\/}
(Lectures on the theory of
finite continuous transformation groups (1902-1903)),
Spoerri, Pisa, 1918.
\bibitem{milnor}
John Milnor,
``Curvatures of Left Invariant Metrics on Lie Groups,"
{\it Adv.\ Math.\/} {\bf 21}, 293 (1976).
\bibitem{jantzen}
R.T. Jantzen,
``The Dynamical Degrees of Freedom in Spatially Homogeneous Cosmology,"
{\it Commun.\ Math.\ Phys.\/} {\bf 64}, 211 (1978).
\end{thebibliography}
\bigskip
{\bf Acknowledgement} The Editors of GRG are grateful to F. de Felice for
his
help in obtaining the publisher's permission for the translation and
reprinting.
\bigskip
\noindent
---Robert Jantzen\\
\phantom{---}Department of Mathematical Sciences, Villanova University,
Villanova, PA 19085 USA
\bigskip
\begin{center}{\bf Brief biography}\\ \end{center}
Born in Parma, Italy on January 18, 1856, Luigi Bianchi began his
mathematics
studies at the Scuola Normale Superiore of Pisa in 1873 and then became a
student of Ulisse Dini and Enrico Batti at the University of Pisa where he
got
his mathematics degree with distinction in 1877, with a dissertation on
applicable (isometric) surfaces. After postgraduate studies in Pisa, Munich
and
then G\"ottingen where he studied with Felix Klein, he returned to Pisa to become
a
professor at the Scuola Normale in 1881 and was appointed as the chair in
projective geometry in 1896. In the same year he became chair of analytic
geometry at the University of Pisa and was later appointed as the director
of
the Scuola Normale Superiore of Pisa in 1918, holding both positions until
his
death in Pisa in 1928 \cite{gillispie,mac}. He had also been an editor of
{\it
Annali di Matematica pura ed applicata\/} and a member of the Accademia
Nazionale dei Lincei.
His mathematical contributions, published in eleven volumes by the Italian
Mathematical Union \cite{opere}, cover a rather wide range of topics. In the
field of Riemannian geometry, he is most well known for his discovery of the
``Bianchi identities" satisfied by the Riemann curvature tensor \cite{bids}
(1902). In 1898 Bianchi published his complete classification of the
isometry
classes of Riemannian 3-manifolds \cite{bianchipaper,bianchibook2}, the more
well known symmetry types categorized by his famous nine types identified by
the Roman numerals I--IX, building upon the theory of continuous groups just
developed by Sophus Lie \cite{lie1,lie2,lie3} (1883--93) and the Killing
equations found by W. Killing (1892) \cite{killing} a few years earlier.
Bianchi played an important role in the generations of mathematicians of the
late 1800's and early 1900's who developed differential and Riemannian
geometry
and transformation group theory and their applications after their
introduction
by Gauss and Riemann and Lie, improved by the tensor analysis methods of
Gregorio Ricci-Curbastro (developer of ``Ricci Calculus" and also a student
of
Dini at the same time as Bianchi) and Tullio Levi-Civita (himself a former
student of Ricci), which in turn influenced the development and birth (1915)
of
the new field of Einstein's general relativity. After Bianchi's death, his
former student Guido Fubini characterized much of Bianchi's work as being a
careful investigation of the many cases which can occur in answering a given
mathematical question \cite{fubini}, certainly a fitting description of his
long article categorizing 3-geometries with symmetry.
\begin{thebibliography}{00}
\setlength{\itemsep}{0pt}
\bibitem{gillispie2}
Charles C. Gillispie, editor,
{\it Dictionary of Scientific Biography\/},
Scribner's Sons, New York, 1970.
\bibitem{mac}
MacTutor History of Mathematics Archive:\\
http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Bianchi.html
\bibitem{opere}
Luigi Bianchi,
{\it Opere\/} (The Collected Works of Luigi Bianchi),
Rome, Edizione Cremonese, 1952.
\bibitem{bids}
Luigi Bianchi,
{\it Rend. Accad. Naz. dei Lincei\/} {\bf 11}, 3 (1902).
\bibitem{bianchipaper}
{\it Sugli spazi a tre dimensioni che ammettono un
gruppo continuo di movimenti\/}
(On 3-Dimensional Spaces Which Admit a Continuous Group of Motions),
{\it Memorie di Matematica e di Fisica della
Societa Italiana delle Scienze\/},
Third Series, Vol.~XI (1898), pp.~267--352;
reprinted in:
{\it Opere\/} (The Collected Works of Luigi Bianchi),
Rome, Edizione Cremonese, 1952, vol.~9, pp.~17--109.
\bibitem{bianchibook2}
Luigi Bianchi, {\it Lezioni sulla teoria dei gruppi continui finiti di trasformazioni\/}
(Lectures on the theory of
finite continuous transformation groups (1902-1903)),
Spoerri, Pisa, 1918.
\bibitem{lie1}
Sophus Lie and Georg Scheffers,
{\it Vorlesungen \"uber Continuierliche Gruppen
mit Geometrischen und Anderen Anwendungen\/}
(Lectures on Continuous Groups with Geometric and Other Applications), Leipzig,
Germany, 1983 [reprinted by Chelsea Pub Co, Bronx, NY 1971].
\bibitem{lie2}
Sophus Lie (with Friedrich Engel), {\it Theorie der Transformationsgruppen\/}
(Theory of Transformation Groups), Volumes 1--3, Liepzig, Germany,
1888,1890,1893 [reprinted by Chelsea Pub Co, Bronx, NY 1970].
\bibitem{lie3}
Lie Groups: History, Frontiers and Applications Volume 1: Sophus Lie's 1880
Transformation Group Paper, translated by Michael Ackermann, comments by Robert
Hermann, Math Sci Press, Brookline, Massachussetts, 1976.
\bibitem{killing}
W. Killing,
``\"Uber die Grundlagen der Geometrie" (On the Foundations of Geometry),
{\it Journ. f\"ur die r. und ang. Math. (Crelle)\/},
{\bf 109}, pp.~121--186 (1892).
\bibitem{fubini}
Guido Fubini,
``Luigi Bianchi e la sua opera scientifica,"
in {\it Annali di matematica pura ed applicata\/} {\bf 62}, pp.~45--81
(1929).
\end{thebibliography}
\begin{flushright}
by Robert Jantzen
\end{flushright}
\end{document}