this book exclusively deals with a special class of
Finsler metricsRandersmetrics．which are defined as the sum of a
Riemannian metric and a l一form．Randers metrics derive from the
research on General Relativity Theory andhavebeen applied in many
areas of the natural sciences．They can also be naturallydeduced as
the solution of the Zermelo navigation problem．The book
providesreaders not only with essential findings on Randers metrics
but also the core ideasand methods which are useful in Finsler
geometry．It will be of significant interestto researchers and
practitioners working in Finsler geometry，even in
differentialgeometry or related natural fields．
Xinyue Cheng is a Professor at the School of Mathematics and
Statistics ofChongqing University of Technology，China．Zhongmin Shen
is a Professor atthe Department ofMathematical Sciences ofIndiana
University Purdue University，TISA
目錄：
Chapter 1 Randers Spaces
1.1 Randers Norms
1.2 Distortion and Volume Form
1.3 Caftan Torsion
1.4 Duality
Bibliography
Chapter 2 Randers Metrics and Geodesics
2.1 Randers Metrics
2.2 Zermelo''s Navigation Problem
2.3 Geodesics
2.4 Randers Metrics of Berwald Type
Bibliography
Chapter 3 Randers Metrics of Isotropic S-Curvature
3.1 S-Curvature
3.2 Isotropic S-Curvature in Terms of a and3
3.3 Isotropic S-Curvature in Terms of h and W
3.4 Examples of Isotropic S-Curvature
3.5 Randers Metrics with Secondary Isotropic
S-Curvature
Bibliography
Chapter 4 Riemann Curvature and Ricci Curvature
4.1 Definitions
4.2 Riemann Curvature of Randers Metrics
4.3 Randers Metrics of Scalar Flag Curvature
Bibliography
Chapter 5 Projective Geometry of Randers Spaces
5.1 Projective Quantities
5.2 Douglas-Randers Metrics
5.3 Weyl-Randers Metrics
5.4 Generalized Douglas Weyl Randers Metrics
Bibliography
Chapter 6 Randers Metrics with Special Riemann
CurvatureProperties
6.1 Ricci-Quadratic Randers Metrics
6.2 Randers Metrics of R-Quadratic Curvature
6.3 Randers Metrics of W-Quadratic Curvature
6.4 Randers Metrics of Sectional Flag Curvature
Bibliography
Chapter 7 Randers Metrics of Weakly Isotroplc Flag
Curvature
7.1 Weak Einstein Randers Metrics
7.2 Randers Metrics of Weakly Isotropic Flag Curvature
7.3 Solutions via Navigation
7.4 Weak Einstein Randers Metrics via Navigation Data
Bibliography
Chapter 8 Projectively Flat Randers Metrics
8.1 Projectively Flat Randers Metrics of Constant Flag
Curvature
8.2 Projectively Flat Randers Metrics of Weakly Isotropic
FlagCurvature
8.3 Projeetively Flat Randers Metrics on S
Bibliography
Chapter 9 Conformal Geometry of Randers Metrics
9.1 Conformally Invariant Spray
9.2 Conformally Flat Randers Metrics
9.3 Conformally Berwaldian Randers Metrics
Bibliography
Chapter i0 Dually Flat Randers Metrics
10.1 Dually Flat Finsler Metrics
10.2 Dually Flat Randers Metrics
10.3 Dually Flat Randers Metrics with Isotropic
S-Curvature
Bibliography
Index
內容試閱：
Chapter 1
Randers Spaces
Randers spaces are nite dimensional vector spaces equipped with
a Randers norm.
Euclidean norm is the most special Randers norm.Roughly
speaking,a Randers norm is a shifted Euclidean norm.If the unit
sphere of a Euclidean norm is called a round sphere,then the unit
sphere of a Randers norm is an ellipsoid.Randers norms are special
Minkowski norms whose unit sphere is a strong convex
hypersurface.
More precise denition is given as follows.
Let V be a nite dimensional vector space.A Minkowski norm on V
is a function
F:V[0;+1 which has the following properties:
a F is C1 on V nf0g;
b F is positively homogeneous of degree one,that
is,F?y=?Fy for any
y 2 V and 0;
c for any y 2 V nf0g,the fundamental form gy on V is an inner
product,where
gyu;v:=
12@2
@s@t ￡F2y+su+tv¤js=t=0:
The pair V;F is called a Minkowski space.A Minkowski norm F is
said to be
reversible if F?y=Fy for y 2 V.
Let V;F be an n-dimensional Minkowski space and feign
i=1 be a basis for V.
View Fy=Fyiei as a function of yi 2 Rn.Put
gijy:=
12@2F2
@yi@yj y:1.1
Then
gyu;v=gijyuivj;u=uiei;v=vjej:1.2
It follows from the homogeneity of F that
Fy=qgijyyiyj;y=yiei:
Lethijy:= FyFyiyj y=gijyFyi yFyj y
andhyu;v:= hijyuivj;u=uiei;v=vjej:
We have
hyu;v=gyu;vFy?2gyy;ugyy;v:
Observe that
hyu;ugyu;uFy?2gyy;ygyu;u=0:
Thus hyu;u0 and equality holds if and only if u=?y for
some ?.hy is called the angular form.
1.1 Randers Norms
First,we consider Euclidean norms.Let Rn denote the standard
vector space of dimension n.The standard Euclidean norm jj on Rn is
dened by
jyj:=vuut
n Xi=1jyij2;y=yi 2 Rn:
Clearly,it is a special Minkowski norm.The pair Rn;jj is
called the standard Euclidean space.More general,let h;i be an
inner product on a vector space V
with a basis feign
i=1.Dene
?y:= phy;yi=qaijyiyj;y=yiei;
where aij:= hei;eji.Clearly,? is a Minkowski norm with
gyu;v=hu;vi inde-
pendent of y 2 V nf0g:? is called a Euclidean norm and the pair
V;? is called a
Euclidean space.It is well-known that all Euclidean spaces with
the same dimension
are linearly isometric to each other.
Now,we introduce Randers norms.Let ?=paijyiyj be a Euclidean
norm on
a vector space V and =biyi be a linear functional on V.Let
Fy:= ?y+y:1.3
It is easy to verify that for any pair of vectors u;v 2 V,
Fu+v=?u+v+u+v
6?u+?v+u+v
=Fu+Fv:
Let 1gu;v denote the inner product determined by ?.Then we
have the following
inequality for any pair of vectors y 6= 0;u 2 V:
1gy;u 6 ?y?u;
and equality holds if and only if u=?y for some ?.For
Fy=?y+y,we
have
gyy;u=
12??+￠2
yiyj yyiuj
=Fyhu +
1gy;u
?y i
6Fyhu+?ui= FyFu;1.4
and equality holds if and only if u=?y for some ?.
Let b:= kk? denote the length of with respect to ?.It is
given by
b=qaijbibj ;
where aij=aij?1.To nd a condition on under which F=?+
is a
Minkowski norm,we compute gij:=
12
[F2]yiyj and obtain the following:
gij=F
? haij+?
F 3bi+yi
3bj+yj
yi?yj
? i;1.5
where yi:= aijyj.
In order to nd the formulas for detgij and gij:= gij?1,we
need the
following lemma:
Lemma 1.1.1 [BaChSh] Let gij and mij be two n ￡ n
symmetric matrices
and c=ci be an n-dimensional vector;which satisfy
gij=mij+?cicj ;
whereis a constant.Then
detgij=1+?c2 detmij:1.6
Assume that mij is positive denite with mij?1=mij and
1+?c2 6= 0.Then
gij is invertible and gij=gij?1 is given by
gij=mij ?
1+?c2 cicj;1.7
where ci=mijcj and c=pmijcicj.
Now,let
mij:= aij+?
F 3bi+yi
mij is a positive denite matrix.Letting ?:= ?=F and ci:=
bi+yi=? in 1.6,
we get by 1.8 that
detmij=detaijh1+?
=detaijh1+?
F 31+2
?
+ b2′i
=detaij
2F++?b2
F:By 1.7,we get
mij =aij ?
?=F
2++?b2=F 3yi
?+ bi′3yj
?+ bj′
=aij ?
?2F++?b2 3yi
?+ bi′3yj
?+ bj′;
where mij:= mij?1.Further,let ?:= ?1 and ci:= yi=?.We
have
1+?mijcicj =1haij ?
?2F++?b2 3yi
?+ bi′3yj
?+ bj′iyi
?yj
?= F2
?2F++?b2:
By 1.51.7,we obtain the following formulas:
detgij=3F
? ′n+1
detaij;1.9
gij=?
Faij ?
F2 biyj+bjyi+b2?+
F3 yiyj:1.10
From the denition of the angular metric tensor,we have the
following formula
for Randers metrics:
hij=FFyiyj=?+ ￠
1aij?yi?yj =?+
3aij ?
yiyj
?2 ′:1.11
Clearly,Fy0 for all y 6= 0 if and only if b
1.Further,gij is positive
denite if and only if b 1 [BaChSh],[BaRo],[Ma1].In
fact,when Fy0,
F":=+ "0
for any 0 6 " 6 1.Let g"
ij:=
12￡F2
" ¤yiyj.By 1.9,we have
detg"
ij=3F"
? ′n+1
detaij0:
Let ?1" 6 ?2" 6 … 6 ?n?1" 6 ?n" denote the eigenvalues
of g"
ij .The
multiplicity of the eigenvalues might change as " changes,but
each eigenvalue ?i"
depends on " continuously.Thus,from ?i00 and detg"
ij0 for 0 6 " 6 1,we
have ?i10.Namely,gij=g1
ij is positive denite.
A Minkowski norm in the form 1.3 is called the Randers
norm.Randers norms
were rst introduced by physicist G.Randers in 1941 from the
standpoint of general
relativity [Ra].
1.2 Distortion and Volume Form
Let V;F be an n-dimensional Minkowski space and feign
i=1 be an arbitrary basis
on V,and fμign
i=1 be the basis for V ¤ dual to feign
i=1.Put
?F:=
VolBn1
Volfyi 2 RnjFyiei 1g
;
where Vol denotes the Euclidean volume and VolBn1 denotes
the Euclidean
volume of the unit ball in Rn.Put
dVF:= ?F μ1 ^ … ^ μn:
It is clear that dVF is well-dened,namely,independent of the
choice of a particular
basis.dVF is called the volume form of F on V.Put
? y:= ln pdetgijy
?F:1.12
It is easy to verify thaty is well-dened.is called the
distortion of F.
If F=paijyiyj is a Euclidean norm,then
Volnyi 2 RnjFyiei 1o=
VolBn1
pdetaij
:Thus
?F=qdetaij:
Note that gijy=aij.We have
? y=ln pdetaij
?F= 0:
Consider a Randers norm F=?+ on an n-dimensional vector space V
with
b:= kk? 1.Let dVF=?F μ1 ^ … ^ μn and dV?=??μ1 ^ … ^ μn
denote the
volume forms of F and ?,respectively.Let feign
i=1 be an orthonormal basis for
V;?.Thus ??=pdetaij=1.We may assume that =by1.Then
-:= fyi 2 RnjFyiei 1g
is a convex body in Rn and ?F=VolBn1=Vol-.- is given
by
1b223y1+b
1b2 ′2
+ 1b2
n Xa=2
ya2 1:
Consider the following coordinate transformation,?:yiui:
u1=1b23y1+b
1b2 ′;ua=p1b2 ya:1.13
? sends - onto the unit ball Bn1 and the Jacobian of
?:yiui is given by
1b2 n+1
2:Then
VolBn1=ZBn1
du1 … dun=Z-
1b2 n+1
2 dy1 … dyn
=1b2 n+1
2 Vol-:
Then
?F=
VolBn1
Vol-
=1b2 n+1
2:Thus for a general base feign
i=1,we have
?F=1b2 n+1
2 ??;??=qdetaij:
Therefore
dVF=1b2 n+1
2 dV?:1.14
Note that
dVF 6 dV?:
The equality holds if and only if b=0 F is a Euclidean
norm.
By 1.9,the distortion of F is given by
?=n+1 lnr1+=?
1b2:1.15
Since j=?j 6 b,we get
n+1 ln
1p1+b
66 n+1 ln
1p1b
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