Characterizations of geometrical mappings under mild hypotheses

Characterizations of geometrical mappings under mild hypotheses

Characterizations of geometrical mappings
under mild hypotheses
By Wen-ling Huang
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Historical Remarks
A mapping of Rn (2 ≤ n < ∞) to itself which preserves the Euclidean distance 1 is already
a Euclidean motion of Rn . No further assumption such as continuity or differentiability are
needed. This theorem has been proved by F.S. Beckman and D.A. Quarles in 1953.
A bijection of four-dimensional Minkowski space M4 which preserves Lorentz-Minkowski
distance 0 is already product of a dilatation and a Lorentz transformation. This is a consequence
of a theorem which A.D. Alexandrov published in 1950 [3].
The theorems of Beckman, Quarles, and Alexandrov established a new geometrical discipline
called Characterizations of geometrical mappings under mild hypotheses. A.D. Alexandrov has
called Theorems of this kind Foundations of space-time geometry when they deal with space-time
geometries.
In the following we give an overview over some results. Many more results may be found
in [7, 8, 9, 27], and in [6, 4, 5, 13]. We start with results on fundamental geometries. Then we
continue with results which are motivated by physical theories such as Alexandrov’s theorem,
and which are based on Lorentzian manifolds, the basis of Einstein’s general theory of relativity.
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Fundamental Geometries
J. Lester proved the following mild-hypothesis characterization for Euclidean motions [26]: A
mapping of Rn , n ≥ 3 to itself, which takes the three edges of a triangle with area 1 to the
edges of a triangle with area 1, is a Euclidean motion. In the case n = 2, however, one obtaines
equiaffine mappings.
Similar theorems hold in the corresponding line geometries [21]. A bijection of the set of lines
of R3 to itself, which obtain the line distance 1 in both directions, is induced by an isometry of
R3 [25]. A bijection of the set of lines of Rn , n > 3, to itself, which takes perpendicular lines to
perpendicular lines and vice versa, is induced by a similarity transformation of Rn [10].
The theorem of Beckman and Quarles is also true in hyperbolic geometry [12]. A mapping
of n-dimensional hyperbolic space (n ≥ 2) which takes hyperbolic lines to lines, such that the
image is not a line, is a hyperbolic motion [15].
L.K. Hua has examined the geometries of matrices [16, 17, 18, 19, 20] and determined for
the several kinds of geometries all the bijections which preserve adjacency in both directions.
Wan [32] posed the question whether it is needed that the bijections preserve adjacency in both
directions. This question has been answered for hermitian, symmetric, and rectangular matrices.
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Theorems for Lorentzian Manifolds
Lorentzian manifolds are the mathematical foundation for Einstein’s general theory of relativity
in physics. Around 1915, ten years after the special theory of relativity was proposed, Einstein
introduced his general relativity. It is the only generally accepted theory of gravity and the
universe in the large. There are no experiments which contradict this theory. Einstein’s famous
field equations involve the Ricci tensor and the metric tensor of a four-dimensional spacetime,
and the energy-momentum tensor which describes the physical world. Thus gravity appears as
a geometrical concept.
The special theory of relativity does not include gravitational effects. Mathematically, it
is the four-dimensional Minkowski space M4 , which is up to isometry the unique complete,
flat, simply connected Lorentzian manifold ([28, p 332]). Two events r1 = (x1 , y1 , z1 , t1 ) and
r2 = (x2 , y2 , z2 , t2 ) in M4 are at distance 0, i.e.,
(x1 − x2 )2 + (y1 − y2 )2 + (z1 − z2 )2 − (t1 − t2 )2 = 0
if, and only if, an unreflected light signal (photon) can travel between r1 and r2 . Every Lorentz
transformation preserves the speed of light.
The physical meaning of Alexandrov’s theorem is that a bijection of M4 which preserves
light speed, is already a Lorentz transformation, up to a dilatation. When Einstein obtained his
Lorentz transformations in special relativity he assumed that these transformations are affine
transformations. He then determined the coefficients of the affine transformations by physical
assumptions.
Apart from Alexandrov-type theorems which deal with distance-zero preserving mappings,
there is another important group of theorems called Zeeman-type theorems.
Zeeman’s theorem [33] characterizes the orthochronous Lorentz transformations with the
fundamental notion of causality. An event r1 = (x1 , y1 , z1 , t1 ) might causally influence another
event r2 = (x2 , y2 , z2 , t2 ) by a material particle, if
r1 < r2
:⇔
(x1 − x2 )2 + (y1 − y2 )2 + (z1 − z2 )2 < (t1 − t2 )2
∧
t1 < t2 .
The theorem states that a bijection of M4 which preserves the < relation in both directions, is
an orthochronous Lorentz transformation up to a dilatation.
S.W. Hawking [14] proved that the conformal mappings between strongly causal space-times
are exactly those homeomorphisms which take null geodesics to null geodesics. W.-l. Huang has
shown that the homeomorphy pre-supposition is not needed if it is aditionally assumed that also
pre-images of null geodesics are null geodesics. [22].
June Lester started [24] to examine Robertson-Walker space-times and proved Alexandrovand und Zeeman-type theorems. Robertson-Walker space-times are Lorentzian manifolds which
satisfy Einstein’s field equations
Rij = λgij + κTij
and which are spatially homogeneous.
The distance 0 preserving bijections for the three-dimensional Einsteinsche cylinder universe
is closely connected to Lie transformations in real Lie geometry, and Alexandrov’s theorem is
closely connected to real Laguerre geometry [7].
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Any bijection from n-dimensional Schwarzschild space-time, n ≥ 3 to itself which takes null
lines to null lines and vice versa, is an isometry [23]. (A null line is the image of a maximal null
geodesic.)
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