Progress in Mathematics: Algebra and Geometry

253 Pages·1971·1.31 MB·English

by  M. S. Tsalenko

FINSLER SPACE> AND THElR GENERALIZATIONS 89
Rl = R2 and F n is a Finsler space of constant scalar curvature
R = Rl (see Yuen-da Wang [555]). In order for an F n admitting of
one family of completely extremal hypersurfaces of constant scal
ar curvature to be aspace of constant scalar curvature, it is nec
essaryand sufficient that the curvature of the space and the curva
ture of the hypersurface be the same (see Yuen-da Wang [555]). If
Rii is the Ricci tensor for the tensor Rhh and if .
(D D 1 i)
2M g'k'V i = �~�k�l� + �~�l�k� + iJdRuk! V dduS' '
where M = F-2R .. vivi, then Fn is called an Einstein-Finsler
space. Any Fins1ier space of constant scalar curvature is an Ein
stein- Finsler space (see Rund [451]). Aspace F is called a
n
space of quasiconstant curvature if
ROJik = ROIOklj - ROlOjlk (n >-3).
In such spaces we can pass a transversal hypersurface through
any !ine element, such that its normal would coincide with the !ine
element (see Rapscak [432]). The scalar
Ko = 1/0ln_1 �~� Ro(x, v) d:J.,
T
where T is the unit tangent vector layer, Ro is the scalar Ricci
curvature, i.e., Ro= gikRik and wn-l is the volume of the layer
relative to the naturalmetric, is called the total curvature of the
space F n. If F n is without conjugate points, then always Ko :s 0
(see Moalla [373]). The lower bound of the are lengths of the curves
of class Cl connecting two points of space F n is called the distance
d between these two points. The space F n is called complete if
every Cauchy sequence converges. For complete Finsler spaces
there holds a Hopf - Rinow theorem analogous to the correspond
ing theorem for Riemann spaces. Aspace F n is complete if and
only if (see Moalla [372, 373]):
1) any geodesie arc can be extended in both directionsj
2) any bounded point set E c F is relatively compact (the
n
closure is compact)j
3) any two points can be connected by a geodesie are whose
length equals the distance between these points.
90 V. 1. BLIZNIKAS
If Ro �~� e2 (e is a eonstant), then in F n tl}ere does not exist a
geodesie are whose minimallength �~� 7r (n - 1) 2" e-1• If F n is eom
plete and Ro �~� e2, then its universal spaee is eompaet and the
fundamental group is finite. If Ro > 0, then the first Betti number
for F n equals zero (see Moalla [371]).
If Xi and yi are veetors, then the quantity (A is a biveetor
formed from Xi and yi)
K ( A) _ RiikhX1YiXkyh
x, v, - IIX1! 21/YJI2-(XY)2 ,
IIxII
where is the length of veetor Xi and (XY) is the sealar prod
uet, is ealled the sealar eurvature in the sense of Akbar-Zadeh
[140]. If K does not depend on A, then Fn is ealled a partially iso
tropie spaee. A spaee F n is partially isotropie for a fixed Une ele
ment (xi, vi) if and only if (n > 2)
R}kl = Ro �(�8�~� glj - �o�~� gkj)'
where Ro is the (eonstant) Rieei eurvature, and in the ease when
Ro �~� 0 it is a spaee of eonstant eurvature in the sense of Berwald.
If �~� Aare veetors (a = 1,2, ... , m -1) defining an (m -1)-dimen-
a .
sional path and an rn-dimensional path together with Vi, then the
quantity
where K/ikh is the eurvature tensor of Rund eonneetion [450], is
ealled the sealar eurvature at the point (xi-) and in the given m
dimensional path. When R is independent of the (m - 1)-dimen-
m
sional path �~� , such F n are ealled (m - 1)-isotropie Finsler spaees
(see Wrona [566]). Spaees F n in whieh the Rieci tensor Kij for the
eurvature tensor K}kh of the Rund eonneetion has the form
are (m - 1)-isotropie and (1 < m < n)
R m-2
m =m(n-l)(n-2) K,
FINSLERSPACES AND THmRGENERALIZATIONS 91
where K is the Ricci curvature (for the tensor K:"h). If F n is a
Finsler space with absolute paralleUsm of line elements, then it is
an (m - l)-isotropic space if and only if (2 < m < n - 1)
2K
K1fkh = (n-I)(n-2) g[/[k (g/Jh]-21nlh]).
and if R is independent of vi, then R is constant (see Wrona
m m
[566-568]).
Moor initiated the research of Finsler spaces with recurrent
curvature [390], te., spaces Fn for which the covariant derivatives
of one of the curvature tensors is proportional to the tensor itself
(the coefficient of proportionality is a covector).
Aspace F n with the metric function F is called the product
of the spaces Fr and F n_ r if F t and F 2 are connected by the relation
F2 = �F�~� + �F�~�.� Such spaces were considered by Varga [538], and he
found the necessary and sufficient conditions for the decomposa
bility of the space F n into two spaces. Aspace Fn is called the
product of two Riemann spaces with quadratic forms PI =aabdxadxb
and p2=apqdxPdxq, if F = f(Pt, P2). Such spaces were investigated
by Hou-Sung Hu [262]. Symmetrie Finsler spaces, i.e., spaces
whose curvature and torsion tensors are covariantly constant, were
investigated by Ting-chia Hsing [111]. Singular Finsler spaces
have been considered by Vagner [28], Zhotikov [56], Kabanov [66],
and Varga [538]. The papers by Varga [530, 531, 535, 543, 544]
and by Barthel [151, 152] are devoted to Minkowski spaces. Varga
[534] found the conditions for the space of Une elements with affine
connection to possess a Finsler metric (also see Rapscak [434,4371).
Papers have begun to appear recently on the geometry of tan
gent bundles of Riemann and Finsler spaces (and also of tangent
bundles of arbitrary fiber spaces). The variety T (R) of tangent
vectors to aRiemann space Rn is called the first-order tangent
bundle of Rn (it is a 2n-dimensional variety). Higher-order tangent
bundles are defined analogously. It turns out that the Finsler met
ric of space F n induces certain algebraic structures of the tangent
bundle for Fn (see Akbar-Zadeh [142, 144], Kurita [332], and Yano
andDavies [5761). The papers by Heil [241], Prakash [423], and
Rizza [442,443] are devoted to complex structures of Finsler spaces.
92 V. I. BLIZNIKAS
Various classes of Finsler spaces (as weIl as certain other
topics) are taken up by Grossman [229], Varga [529], Vasil'ev [34],
Kawaguchi [282-284], Süss [499], Kikuchi [317], Laugwitz [336-340],
Misra [362-366], Moor [379,384-387,391,393,396,398], Nasu [402,
403], Sen [459, 460], Rashevskii [106,107,438-440], Shirokov [133],
Soos [474,476], Rapscak [435,436], Sasayama [455], Shamihoke [466],
Tachibana [503], Theodoresco [511-513], and Otsuki [416].
§ 4. Motions and Transformations
Transformations which preserve the metric are called the
motions of the Finsler space Fn. The general theorem on motions
was proven by Laptev [82] who was the first to apply the concept of
the Lie derivative to the search for the motions and other trans
formations of Finsler spaces. If the motion group of a Finsler
space F n is of maximal possible order, then the space is Euclidean
(see Yano [571]). If the space F admits of an r-parameter motion
n
group and r > (n (n -1)/2) + 1, then F n is aspace of constant curva
ture (see Yuen-da Wang [555]). Chao-Hao Ku proved [330] that a
space F n (n > 4, n ;t 8) admits of a motion group of order r =
(n(n-1)/2) +1ifthemetric function Fis positive definite and has
the form (a = 2, ... , n):
or
Tashiro [509] found certain classes of spaces Fn admitting of a mo
tion group of the same order. Maurin [357] considered the motion
groups of a generalized Finsler space F n (when the indicatrix is a
star-shaped hypersurface). The space of variables (xi, Pi)' where
Pi = a In F / avi, is called a phase space of space F n. The one-term
group of canonical transformations of a 2n-dimensional phase
space generates a vector field to which there uniquely corresponds
the Hamilton function H (x, p) defining the figuratrix and the indi
catrix of some Finsler space whose geodesic lines correspond to
the trajectories of this group in the phase space .. The Finsler met
ric also generates a one-term transformation group in the phase
space.
FINSLER SPACES AND THElRGENERALIZATIONS 93
Infinitesimal transformations defined by the vector field
�~� i (x) are called homothetic transformations of a Finsler space if
(LI; is the symbol for the Lie derivative)
L;glj =egij (e> 0, c = const).
If c = 0, the homothetic transformations coincide with motions. If
the space F n admits of an (r + l)-parameter group Hr+1 of homo
thetic transformations, then this group always has an r-parameter
motion group G of space F as a normal divisor. This result is
r n
due to 806s [475]; he proved the nonexistence of Finsler spaces
Fn (n �~� 4) with a homothety group Hr, where in(n + 1) > r > in·
(n - 1) + 2. Homothetic transformations always preserve the tor
sion and curvature tensors of the Finsler space Fn• Akbar-Zadeh
proved [145] that Ü Fn is a locally-unreduced analytic variety, then
every infinitesimal transformation which preserves the torsion and
curvature tensors (as weIl as the covariant derivatives of these
tensors) is also an infinitesimal homothety.
Transformations preserving affine connection are called af
fine motions of space F n. Matsumoto proved [352, 353] that a
space F admits of a maximal group of affine motions if and only
n
if all torsion and curvature tensors of the given connection equal
zero. 806s found the connection between homothetic and affine mo
tions of a Finsler space. Homothety transformations do not exist
in a Finsler space of constant curvature. In order for an r-param
eter group G of affine motions of space F to have at least a one
r n
dimensional homothety subgroup it is necessary and sufficient that
where �~�a� is the basis of the Lie algebra of group Gr• If s < r + 1,
then the subgroup of homothetic transformations has the dimension
r + 1 - s (see So6s [472, 4731). Homothetic transformations were
studied by Shanks [468] and Yano [570, 571]. If the order öf the
homothetic transformations of space F n is maximal, then the space
F n is Euclidean (see Hiramatsu [245, 2461). 806s [472] found the
condition for the affine motion group of a Finsler space F n to con
tain the motion group of space F n as a nontri vial subgroup. Homo
thetic motions of two-dimensional Finsler spaces have been inves
tigated in detail by Chetyrkina [125].
94 V. I. BLIZNIKAS
Finsler spaces for which the tensors �A�~�.�~� and RfkZ - AJmRökz
are covariantly constant are called quasisymmetric spaces (see
Soos [474]). Any quasisymmetric space F admits of at least a
n
one-parameter affine motion group. If the matrices X"h = RJkh
11 11
form an irreducible system of linear transformations, then the geo
desics coincide with the autoparallel curves (see Teodorescu [514]).
Transformations which preserve the form ' F iml are called
dilations of the space Fn• Kurita [331,332] proved that the volume
metric of a Finsler space (the volume of the hyperflat element dual
to a Une element) is invariant relative to dilations.
Conformal transformations of Finsler metrics have been con
sidered by Nazim [404], Hombu [251, 252], Misra [362,363], and
Rund [450]. Akbar-Zadeh [141] gave the name conformal motions
of a Finsler space F n also to the infinitesimal transformations for
which
2
Ls (g- i! gz) = 0 (g = det 11 glj 11).
and found the necessary and sufficient condition for the infinitesi
mal transformations of a compact space F n to be conformal mo
tions. Various transformations of Finsler spaces have been treat
ed by Vagner [19], Kagan [70, 71], Akbar-Zadeh [144, 145], Kobay
ashi [322,323], Hokari [247], Knebelmann [320,321], Kondrat'ev
[74, 75], Muto and Yano [400], and Ohkubo [414]. Earlier papers on
motions of Finsler spaces have been covered in Yano's monograph
[571].
In a Finsler space F n there exist co ordinate systems in
which the components of the symmetric part of an object of affine
connection equal zero (an analog of the normal coordinates in a
Riemann space). Such coordinate systems (and also the theory of
the invariants of space Fn) have been considered by Bucur [175],
Busemann [176,177], Davies [186], Moor [392,394], Rapscak [427-
430], and Varga [537].
Various transformations (such as projective, conformal, geo
desic) of connection objects, as weIl as of the object Hi of space
F n' have been analyzed by Hsien-chu Wei [40], Golab [224, 225],
Haimovici [232-237], Hombu [251-253], Misra [362,363], Moor
[389], Sinha [470].
FINSLER SPACES AND THEIR GENERALIZATIONS 95
§ 5. Immersions of Finsler Spaces
The space F n is a fiber space (the layers are isomorphie
Minkowski spaces). The set of local indicatrices of a Finsler
space forms the global indicatrix of this space. We say that the
space F has been embedded in an affine space ANwith a vector
n
metric, if in ANthere exists an n-dimensional surface P n such
that the field, given in AN' of hypersurfaces (they are defined by
the vector metric) under an interseetion with the tangent planes of
surface P forms on the latter a field of local (n - l)-dimensional
n
surfaces (they are taken as the local indicatrices of a new Finsler
space with base Pn) relative to which the surface turns into a Fins
ler space isometrie to the space F n. The immersion is said to be
regular if the surface P is tangentially nonsingular , and is said to
n
be singular if the surface P is singular, Le., the set of its tangent
n
planes depends on p parameters (p< n - 1). In 1949, Vagner
proved that an n-dimensional Finsler space can be regularly em
bedded (locally) in a 2n-dimensional Minkowski space, Le., in a
2n-dimensional affine space with a vector metric. Other proofs of
Vagner's theorem were obtained by Ingarden [267, 268] and by
Chao-Hao Ku [42] who proved that every n-dimensional Finsler
space of class Cf (r 2) is regularly immersed wholly in a 4n
2:
dimensional Minkowski space. Singular immersions of two-dimen
sional Finsler spaces in three-dimensional affine spaces were con
sidered by Kagan [70] who proved that F admits of an imbedding
2
in A with a vector metric in the form of a (cyclic conical or tan
3
gential) surface if and only if the metric function F has the form
(in special coordinates)
By an immersion of a Finsler space in aRiemann space we
mean a mapping under which to the line elements of the Finsler
space there correspond points of the Riemann space RN (N 2: 2n-1)
such that, in the tangent spaces E of the Riemann space RNwe
N
can pick out an n-dimensional plane E and a path E in this plane
n
such that the induced metric of this plane and the projection of an
infinitely-close n-dimensional plane Eri (the projection is estab
lished by means of the affine connection of the Riemann space RN)
coincide with the metric and with the parallel displacement of the
immersed Finsler space. If the first n vectors Ei of the Euclidean
tangent space E2n-1 for R2n- 1 have been embedded in the plane En
96 V. I. BLIZNIKAS
and if the veetor E has been eombined with the path E, then the
1
immersion equations have the form (i. j = 1, 2 ••••• n; a = n + 1 .....
2n -1; I. J = 1. 2 .... , 2n - 1)
(26)
where the pfaffian forms nI and n} are eonneeted by the strueture
equations of the Riemann spaee R2n- 1 with torsion-free eonneetion.
Deike [191] proved that a spaee Fn with Cartan eonneetion ean be
represented as an n-dimensional holonomie surfaee of aRiemann
spaee if �A�~�k� == 0 (mod na). Thus. a Finsler spaee with Cartan eon
neetion ean be immersed as a holonomie surfaee Rn of the spaee
R2n-1 if and only if the system TI i = O. DII i = O. where lli =
Ajk(J)i 1\ (J)k, is in involution. Galvani [214] proved that this system is
always in involution when n = 2. It turns out that any Finsler
spaee ean be immersed as an n-dimensional nonholonomie surfaee
of a (2n - l)-dimensional Riemann spaee with torsion (see Deike
[192]). It should be noted that a Finsler spaee with Berwald eon
neetion ean always be immersed in a (2n - l)-dimensional Riemann
spaee (as an n-dimensional holonomie surfaee). Any two-dimen
sional Finsler spaee with Cartan eonneetion ean be embedded (as a
three-dimensional surfaee) in a five-dimensional Euelidean spaee
(see Galvani [215]) (the arbitrariness of the embedding is four fune
tions of one argument). while a two-dimensional Finsler spaee with
Cartan eonneetion and with absolute parallelism of the support line
elements ean be embedded in a three-dimensional Euelidean spaee
(the arbitrariness of the embedding is two funetions of one argu
ment). The immersions of two-dimensional Finsler spaees were
investigated in detail by Galvani [212-2171.
§ 6. Metrie Spaees of Line Elements
Various generalizations of Finsler spaees exist and, more
over, in some generalizations the metrie function (taken as a funda
mental differential-geometrie objeet) is replaeed by another differ
ential-geometrie objeet whose strueture eoineides with the strue
ture of a eertain differential-geometrie objeet spanned by the dif
ferential eontinuations of the metrie funetion F. while in other gen
eralizations the fundamental objeet (the function F) does not ehange
(in the sense of the strueture) but another fiber spaee is taken. The
generalizations of Finsler spaees (in the seeond sense) will be eon
sidered in other seetions.
FINSLER SPACES AND THEIR GENERALIZATIONS 97
The spaee �L�~�)� of line elements, with a nondegenerate sym
�m�e�t�r�i�~� tensor field �~�j�:�
(27)
is ealled ametrie spaee �~�"� of line elements. It is elear that with
�~�"� we ean alway!> �~�s�s�o�e�i�a�t�e� a Finsler space Fn with the metrie
function F2 = gij vlvJ• The fundamental differenee between the
spaee 6'" and the spaee F n is that gijk �~� �~�j� (in the Finsler
spaee the eorresponding tensor is symmetrie). The geometrie
meaning of this eondition is that in the spaee �~�"� the notions of
transversality and orthogonality are different (in the Finsler spaee
these notions are identieal). The spaees 6'" under various names
(for example, Finsler-Riemann systems, Finsler spaees with non
potential metrie, ete.), are eonsidered by Ohkubo [408], Akbar
Zadeh [138, 140, 142, 143], Moor [377, 378, 380, 381], Freeman
[208, 209], Bliznikas [2-4, 6, 8], Vanstone [528], Prakash [424]. and
�S�h�a�m�i�h�o�~�e� [467]. The geometry of the spaees ff" is fundamental
ly similar to the geometry of the spaees F n' only the nonidentity
of the notions of transversality and orthogonality (orthogonality in
the sense of the metrie of the assoeiated Finsler spaee) shows o.p
in the faet that the usual eoneepts of Finsler geometry are doubled
in number (for example, for these spaees there are more analogs
of the Sehur theorem, the theory of eurves and surfaees is enriehed
by new eurvatures, e te.) •
Spaees of line elements with an asymmetrie tensor hij �~� hjio
Le., bilinear metrie spaee of line elements, have also been eon
sidered (see Bliznikas [5, 7], Shapukov [127-129], Prakash [423],
Moor [395], and Shamihoke [462]). Important results in the theory
of surfaees of a bilinear metrie spaee of line elements were ob
tained by Shapukov [127-129]. Kähler-Finsler spaees were ana
lyzed by Prakash [4231.
Finsler geometry is applieable not only in the ealeulus of
variations (Vagner and his students) but also in physies. Not only
Riemann spaees but also Finsler spaees are taken as a geometrie
model for some phenomenon or other in modern physies, in par
tieular, in unified field theory (see Horvath [254, 255]). For ex
ample, Horvath proved that the Stephenson-Kilmister unified
theory of gravitation and eleetromagnetism ean be formulated in
the language of Finsler spaee eoneepts if as the fundamental metrie
98 V. 1. BLIZNIKAS
function of this space we take the Lagrangian in the Stephenson
Kilmister theory for the test particle which moves under the action
of a unified electromagnetic and gravitational field. A pair of
points is the fundamental element of the model space in the bilocal
theory of Yukawa. Horvath and Moor [256, 257, 258] proved that
the Yukawa space can be mapped onto a variety of line elements,
whose metric structure is established by means of a twice covari
ant symmetrie tensor which is not potential (the metric tensor of
a Finsler space is potential). The nonpotentiality of the metric
characterizes the anisotropy of the space, the concept of which is
of significant value in physical theories.
Applications of Finsler geometry to physics, mechanics, and
optics have been discussed by Vagner [23], Eliopoulos [200, 202],
Fujinaka [210], Segre [458], Hosokawa [260], Horvath [256], Klein
[319], Kondo [324], Lichnerowicz [343-346], Pihl [421], Rund [444],
Suzuki [500], Uehara [525], and Wundheiler [569]. The fundamental
aspects of the integral ·geometry of the space F n are contained in
Blaschke [168] and Owens [420].
§ 7. Cartan Space
The pair consisting of the point xi and the covector ui is
called a hyperflat element. The structure equations of the space
�L�~� of hyperflat elements have the form:
Doi=wk 1\ �w�~� •
+w
�D�a�i�=�W�~� 1\8k kI\8ik• (28)
where ei == dUi - �U�k�W�~� and eij == -WfjUL' If on �L�~� we are given a
scalar field defined by the function F (the fundamental metric func
tion):
(29)
then the space �L�~� is called a Cartan space C (see Cartan [180]).
n
This class of spaces, introduced by Cartan, were named Cartan
spaces at Berwald's suggestion [162-164, 166].
Thesecond differential extension of Eq. (29) includes the
metric tensor and the formula for such an inclusion (in a holonom
ic frame) has the form: