CHAPTER-III
__________________________________________________
SOME CURVES IN HERMITIAN AND TACHIBANA
HYPERSURFACES
1.
INTRODUCTION:
Kronecker [53] at the very beginning of his research , has investigated the curvature of a hypersurface of the n-dimensional Euclidean
space, to verify the existence and the leading features of the lines of curvature as well as the generalization of Meunier's theorem and various other
results. Ricci [104] has published a crucial investigation of the properties
of an orthogonal ennuples in a Riemannian manifold, in which he has used
the coefficients of rotation, which are now universally known as "Ricci's
coefficients". Moreover, Ricci explored that for a given congruence of
curves in a Riemannian manifold Vn , there is a special set of (n-1)-mutually
orthogonal congruences, related to it in particular intimate manner. He
called these congruences canonical with regard to the given congruence.
The congruences of geodesics, normal congruences and n-ply orthogonal
systems of hypersurfaces were also examined. Weatherburn ([192] and
[195]) has described some special features of hypersurfaces, the hypersur-
Chapter-III.nb
88
face in Euclidean space of constant curvature and produced the theory of
certain quadratic hypersurfaces in Riemannian space. Further, Bihari and
Saxena [10] have defined and studied the hyper-Darboux lines of a Kaehler
manifold and several results have been obtained by them. Hypernormal
curves on a Riemannian hypersurfaces and Finsler space have been studied
by Singh ([152] and [155]) and various theorems have been derived. Also,
some properties of such curves have been investigated by Semin ([110] and
[111]). These curves have been generalized to produce the hyper-D lines
(i.e., hyper-Darboux lines) of a hypersurface by Pravanovitch [93]. Furthermore, Singh [154] has analyzed some concepts of hyper-D lines of order h
in Riemannian subspaces and some properties of union hyper-D lines have
also been investigated by him. Springer [164] has characterized the theory
of union curves of a Riemannian hypersurface and various results have
been obtained by him. Further, Singh [157] has studied special, union and
hyper-asymptotic curves of a Riemannian hypersurface and several results
have been obtained therein. Also, Mishra [68] has discussed various proper ties of union curves in a subspace of Riemannian space. Moreover, Tsagas
[182] has studied special curves of a hypersurface of Riemannian space
and a number of significant facts concerning to such curves, have been
mentioned. Recently, Singh and Kothari [134] have defined and studied
special, union and hyper-asymptotic curves of a Tachibana hypersurface
and several theorems have been proved by them. Later, Singh and Thapliyal [133] have characterized union and hyper-asymptotic curves of a
Tachibana hypersurface and several results have been obtained by them.
Chapter-III.nb
89
The present chapter is devoted to the study of some curves in Hermitian and Tachibana hypersurfaces. We have defined and studied the hypernormal curves on the Hermitian hypersurfaces and several properties of
such curves have been discussed. Also, the equations representing a hypernormal curve of order one are obtained. Further, union and hyper-asymptotic curves and some special categories of such curves have been investigated. Moreover, the nature and effects of such special curves in Hermitian
manifolds and Hermitian hypersurfaces have been considered and several
theorems have been investigated. Furthermore, the study of hyper-Darboux
lines in the Tachibana hypersurfaces has been made and some properties of
hyper-D lines have also been investigated. Further, the necessary and sufficient conditions for the curvature congruences to be the hyper-D lines have
been derived therein.
We, now, firstly define Hermitian spaces and outline some basic
formulae, which are the necessary pre-requisites to study such spaces.
Let us consider that there is given a self-conjugate positive definite
Riemannian metric:
(1.1)
ds2 = gij dzi dz j
in the complex manifold Cn .
If the fundamental metric tensor gij is hybrid, then we call such a
metric, a 'Hermite metric' and the complex manifold Cn , equipped with this
metric, is called a Hermitian manifold (Yano & Bochner [205]). We call it
as Hnc - space. We shall always assume the self-adjointness of the indices
in an Hnc - space.
Chapter-III.nb
90
Since, the fundamental metric tensor gij is hybrid, therefore its
covariant components will satisfy the relation:
(1.2)
g ij =
or,
(1.3)
i 0
kg
-
lm
gl m y
-
0 {
,
j
Fhi Fk g kh = gij ,
where Fhi is an almost complex structure field.
On account of the fact that
(1.4)
g lm
=
--
gl m
=0
and
-
(1.5)
g
lm
=g
-
ml
-
=g
lm
-
=g
ml
and making use of the formal definition of Christoffel's symbols, we nay
have the following expressions:
(1.6)
Glm n = Gln m =
(1.6*)
Glm n- = Gl-n m =
1
2
gl rI
-
∑gn r-
∑ zm
1 l r ∑gm r
g
I
2
∑ zn
+
-
∑gm r∑ zn
∑gm n-
∑ zr
M,
M
and
(1.6**)
Glm- -n = Gl-n m- = 0.
The rest components of the Christoffel's symbols are given by the
self-conjugacy of the above three equations.
Now, if a Hermite metric given by equation (1.1), satisfies the relation
(1.7)
Glm n- = Gl-n m = 0,
we call the metric a Kaehler metric and the complex manifold equipped
with this metric is called a Kaehler space (E. Kaehler [48], J.A. Schouten
Chapter-III.nb
91
[107], [108], J.A. Schouten and D. Van Dentzig [109]). We shall denote
such a manifold by Knc .
On account of the above known results, we now, have some more
basic results on subspaces and hypersurfaces.
SPACE OF N-DIMENSIONS, VARIETIES,
DIRECTION AT A POINT AND HYPERSURFACES
A set of n-independent variables x1 , x2 , ...., xn may be regarded as
the co-ordinates of a current point in an n-dimensional space Vn , in the
sense that each set of values of variables defines a point of Vn .
The
totality of points corresponding to values of variables lying between certain specified limits constitutes a region of Vn .
The aggregation of points of Vn , whose co-ordinates may be
expressed as a function of single parameter t, is called n-dimensional space
curve, or a curve in n-dimensional space.
Thus the equations
xi = xi HtL, for i=1, 2, ...., n
define a curve of Vn . Also, a family of curves, one of which passes through
each point of Vn , is called a congruence of curves. Further, the points of
Vn , whose co-ordinates are expressible as functions of two independent
parameters t and s, constitute a surface in Vn .
Equations of the form:
Chapter-III.nb
92
xi = xi Hs, tL, (i=1, 2, ...., n)
thus define a surface. The totality of points, whose co-ordinates are express ible as functions of k independent parameters is called a variety or subspace of Vn of k dimensions, and is denoted by Vk . Any such variety is said
to be immersed in Vn . Now, if k=n-1, Vk is called the hypersurface of Vn .
An equation of the form:
fHx1 , x2 , ...., xn L = 0,
determines a hypersurface. And if c is any arbitrary constant,
fHx1 , x2 , ...., xn L = c
or, more generally, fHxi L = c represents a family of hypersurface and each
value of c determines a unique hypersurface. Hence, hypersurface is a
particular curve of (n-1)-dimensions in Vn .
Suppose that P be a point of Vn , whose co-ordinates are
xi , Hi = 1, 2, ...., nL, and Q any neighbouring point of Vn , whose co-ordi-
nates are xi + dxi , then the differential dxi are called the components of the
infinitesimal displacement PQ in the system of co-ordinates xi . We call that
each displacement determines a direction in Vn at P and the directions of
infinitesimal displacements dxi and dxi at P are regarded as being the same,
if the two differentials are in proportion, that is to say, if
dx1
dx1
=
dx2
dx2
=. ... =
dxn
dxn
.
DEFINITION (1.1): SUBSPACES OF A
RIEMANNIAN MANIFOLD.
Let Vm be a Riemannian space of dimension m, referred to an allow-
Chapter-III.nb
93
able co-ordinate system ya , (a=1, 2, ...,m) and possessing the metric
aa b dya dy b . Now, the points of Vm , whose co-ordinates are expressible as
functions of n-independent variables xi , Hn mL, are said to constitute a Vn
immersed in Vm . Let the metric for Vn be denoted by gij dxi dx j . If xi and
xi + dxi are the adjacent points of Vn , whose co-ordinates in y's are ya and
ya + dya , then we may have
dya =
(1.8)
∑ya
∑xi
dxi , for all (i=1, 2, ...., n) and (a=1, 2,...., m).
Also, since the length ds of the element of an arc connecting the two
points of any space is same, whether it is calculated with respect to Vn or
Vm , it follows that
∑ya
∑xi
gij dxi dx j = aab dya dy b = aab
∑y b
∑x j
dxi dx j .
And as the above relation holds for all values of the differentials, we have
gij = a b
(1.9)
∑ya
∑xi
∑y b
∑x j
.
Thus, the metric of the subspace is determined by the equations
defining the subspaces.
Let dya and dy b be the components of two elementary vectors at a
point of Vn in y's and let dxi , dxi be the corresponding components x's, then
their inclination f, calculated with respect to Vm is given by
Cosf =
" Ha
ab
Haab dya dy b L
dya dy b L Haab dya dy b L
.
By virtue of equations (1.8) and (1.9), the last expression is equivalent to
Cosf =
Hgij dxi dx j L
" Hg dxi dx j L Hg dxi dx j L
ij
ij
,
which shows that f is also the inclination of the vectors calculated with
respect to Vn .
Chapter-III.nb
94
DEFINITION (1.2): HYPER-D-LINES.
A Darboux line (or, D-line) on the surface of 3-dimensional Euclidean space is a curve, for which the osculating sphere at each point is the
tangent to the surface and a hyper-Darboux line on the surface on n-dimensional Euclidean space, is a curve for which the osculating sphere at each
point of the hypersurface of the Euclidean space is the tangent to the hypersurface.
EXPRESSION FOR THE TENSOR DERIVATIVE
OF THE UNIT NORMAL AND THE DERIVED
VECTOR
We, now, discuss the generalized formulae for the tensor derivatives of the unit normal N a with respect to x's and the derived vector of N
with respect to the enveloping Riemannian space Vm , for any direction in
the hypersurface.
The unit normal vector N a is a contravariant vector in the y's,
whose tensor derivative with respect to the x's, is given by:
(1.10)
N a, i =
∑N a
∑xi
+ 9 a = N b yd, i .
bd
We may obtain an expression for this in terms of the quantities Wij where
Wij 's are the components of symmetric covariant tensor of the second order
in the x's. Taking the tensor derivative of each side of the well known
expressions
(1.11)
aab N b ya, i = 0 , Hi = 1, 2, ... .., mL
Chapter-III.nb
95
and
(1.12)
aab N a N b = 1 (Weatherburn [192]),
we get
(1.13)
aab N b N,ai = 0,
which shows that N,ai regarded as a vector in the y's, is orthogonal to the
normal and thereby tangential to the hypersurface. It may be expressed
linearly in terms of n-independent vectors ya, i , so that
(1.14)
N,ai = Aki ya, k .
The coefficient Aki can be determined as follows. Taking the derivative of
equation (1.11) with respect to x's, we have
(1.14*)
b
aab ya, ij N b + aab ya, i N, j = 0.
Now, substituting from the following known expression:
(1.15)
Rhijk =
1
2
I ∑x∑ ig∑xhk j +
2
∑2 gij
∑xh ∑xk
-
∑2 ghj
∑xi ∑xk
+gab 9 b = 9 a = - gab 9 b = 9 a =
hk
i j
h j
-
∑2 gij
∑xh ∑x j
M+
ik
and equation (1.14) in the expression (1.14*), we get
Wij = -aab ya, i y,bk Akj = -gik Akj .
Multiplying by g ih and summing over i, we get
(1.16)
Wij g ih = -Ahj .
Thus, equation (1.14), on interchanging the dummy indices, becomes
(1.17)
N,ai = -Wij g jk ya, k .
The equation (1.17) is the required expression for the tensor deriva-
tive of N a . The derived vector of N for the direction of any unit vector ei
in the hypersurface is therefore given by
(1.18)
N,ai ei = ei Wij g jk ya, k .
Chapter-III.nb
96
Now, let us consider an analytic Hermitian hypersurface Hnc of the
embedding
IV a ,
-
Va
Mª
or,
JV 1 ,
enveloping
-
V 2,
V n,
....,
-
V 1,
V 2,
space
c
Hn+1
.
If
...., V n N denote the co-ordinates of
-
a point in Hnc , then the equation of the Hermitian hypersurface (i.e.,
Hnc - hypersurface) may be written in the form:
zi = zi HV a L ; z i = z i IV a M.
-
(1.19)
-
-
Suppose that ga -b is the fundamental metric tensor, then
-
(1.20)
ga b =
-
gi j Bia
-
j
B- ,
b
where
Bia
∑zi
∑ va
=
-
j
; B =
-
b
-
∑ zj
∑ vb
.
Since Yano [198] has shown that an analytic hypersurface of a
Kaehlerian manifold is also a Kaehlerian manifold, likewise due to the
same reason an analytic hypersurface of a Hermitian manifold will be Hermitian manifold.
Hence, if IN i , N i M be the components of unit normal vector to the hypersur-
face, then we have
-
(1.21)
2 gi -j N i N j = 1
and
-
(1.22)
j
-
gi j N B - = 0 ; gi j N j Bia = 0.
-
i
-
b
Now, if Iqi , q i M and IPa , pa M are the components of first curvature
-
-
c
vector with respect to Hn+1
and Hnc respectively, then we have from Bihari
and Saxena [10]:
(1.23)
and
qi = Bia pa + Kn N i
Chapter-III.nb
97
-
-
-
-
q i = Ba-i pa + K n N i ,
(1.24)
where the normal curvature HKn , Kn L of the hypersurface is given by
M I dV
M,
Kn = WabI dV
ds
ds
a
b
a
Wa- -b J dV
ds
-
Kn =
NJ
-
dV b
ds
N,
and
-
Bia, b
i
= Wab N ;
B i-ab
-
= Wab N i .
--
where comma (,) followed by an index denotes the operator of covariant
differentiation with respect to the metric tensor gij of Riemannian space.
IWab, Wa- -b M are the components of symmetric covariant tensor of the second order in the x's to the hypersurface. It is obvious from the fact that the
functions Bia, b are of this nature.
Now, let us consider a curve C : zi = zi HsL ; z i = z i HsL (where s ia
-
real) of
Hnc .
The components
dzi
ds
Jor,
-
dz i
ds
N and
dVa
ds
Jor,
-
-
dVa
ds
N of the unit
c
and the hypertangent vector of C with respect to the enveloping space Hn+1
surface Hnc are related by
dzi
ds
(1.25)
-
dz i
ds
(1.26)
dVa
ds
= Bia
-
,
-
dVa
ds
= Ba-i
.
Also, the unit vector Ix , x M orthogonal to J
-
i
(1.27)
gi -j
dzi
ds
i
-
-
x j + g-i j
dz i
ds
dzi
ds
-
,
dz i
ds
N is given by
xj = 0
and
(1.28)
2 gi -j
dzi
ds
-
xi x j = 1.
We shall, now study about hyper-normal curves on a Hermitian
hypersurfaces.
Chapter-III.nb
2.
98
HYPERNORMAL CURVES ON A HERMITIAN
HYPERSURFACE
In this section, we will study the curves on a Hermitian hypersurface and various properties of such particular curves of (n-1) dimensions
and several particular cases will be discussed.
Assuming that d /ds is the usual covariant differentiation along a
curve C :
zi
=
zi HsL ;
= z i HsL of the Hnc , where C is non-geodesic and
-
-
zi
non-asymptotic curve, then we have the first two Serret-Frenet's formulae:
dhiH0L
ds
(2.1)
-
i
dhH1L
ds
= KH1L hiH1L and
= -K H1L hiH0L + KH2L hiH2L
together with their complex conjugates given by
-
i
dhH0L
ds
(2.1*)
-
-
=
i
K H1L hH1L
i
dhH1L
ds
and
-
-
i
i
= -K H1L hH0L
+ KH2L hH2L
,
where
9hiH0L I
ª
dzi
ds
M,
i
hH0L
J
-
-
ª
dz i
ds
N=,
9hiH1L ,
i
hH1L
=
-
and
9hiH2L ,
i
hH2L
= are the
-
components of unit tangent, unit principal normal vector and the unit first
binormal vectors respectively, while HKH1L , K H1L L, HKH2L, K H2L L are the first
and second curvatures of the curve C respectively.
Let us consider a congruence of curves given by unit vector field
c
, there passes exactly
l** = Il , l i M, such that through each point of Hn+1
i
-
one curve of congruence. At the point of the hypersurface, we have
(2.2)
li = ta Bia + c N i
and its conjugate, where
sents a unit vector, we get
ta
and c are the parameters. Since
Ili ,
l i M repre-
Chapter-III.nb
99
-
2 gi -j li l j = 1
and it follows by using (1.21), (1.22) and (2.2) that
-
2 ga b
-
ta t b
= 1 - » c »2 .
-
Further, assuming that q is the angle between the unit vectors l** and N ,
we have
c=Cosq and 2 ga -b ta t b = Sin2 q = 1 - » c »2 .
-
Now, a curve of the hypersurface will be called a hypernormal
curve of order h, (h=1, 2,...., n-2), relative to the congruence l** , if the
variety spanned by the first curvature vector and binormal vector
i
IhiHh+1L , hHh+1L
M of order h contains l** .
-
For a hypernormal curve of order one, we have
(2.4)
li = gqi + n hiH2L
Thus, the equations (1.23), (1.17) and (2.1) yield
(2.5)
dqi
ds
=
dpa
ds
- KHnL Wgy g1êa
µ
dVa
ds
+
dKHnL
ds
Bia + IWab pa µ
dVy
ds
M.
Similarly, we may obtain the conjugate of the above expression.
Eliminating hiH2L from (2.4) and (2.5) and making use of the equations
(1.23), (2.2) and (2.3), we find
(2.6a)
and
(2.6b)
where
(2.6c)
a
ta = gpa wA dpds - KHnL Wgy g1êa
- 8dHlog KH1L ê dsL< pa E
Cosq = gKHnL + wIWab pa
µ 8dHlog KH1L ê dsL<,
w = n ê KH1L .KH2L
dV b
ds
dVy
ds
+
2
+ KH1L
dKH1L
ds
dVy
ds
M - KHnL
Chapter-III.nb
100
and their conjugates.
Using the first two Serret-Frenet's formulae with respect to Hnc , we
get
pa =
dVa
ds
a
= KH1L xH1L
and
a
dxH1L
ds
= -KH1L
dVa
ds
a
+ KH2L xH2L
and their conjugates, we have
dpa
ds
(2.7)
2
= -KH1L
+ d 9log
dVa
ds
a
µ xH2L
KH1L
ds
= pa + KH1L KH2L µ
a
a
and its conjugate, where IxH2L
, xH2L
M is the first binormal vector of the curve
-
with respect to hypersurface Hnc .
On substituting this value of I dpds M in the equation (2.6a), we get
a
-
a
a
t = gp
(2.8)
a
2 dV
+ wA9KHnL
ds
µ
dVy
ds
pa = + K
E
+ 9dIlog I KH1L
* M ë ds= µ
K
a
H1L KH2L xH2L
H1L
KHnL Wgy g 1êa
and its conjugate may be obtained in the same way. In the above relation,
we have used
*2 = K 2 + K 2
KH1L
H1L
HnL
(2.9)
for simplification.
Since l** is a unit vector field, we have
(2.10)
2 + K 2 K 2 w2 ,
1 = g2 KH1L
H1L H2L
in the light of equations (2.1), (2.4) and (2.6c). The elimination of g and w,
from (2.6b), (2.8) and (2.10), will yield the equation for hypernormal curve
of order one.
Chapter-III.nb
101
From the equation (2.8), we have
-
b
ga b ta dV
ds
-
= 0;
-
-
b
ga b ta dV
ds
= 0,
- -
which proves the following proposition:
A necessary condition that a curve be hypernormal (of the order
unity) relative to the congruence l** is that the tangential component of
the hypersurface of the vector field l** is orthogonal to the curve.
After defining
Cosa =
Jg
2 $ Ig
ta xH1L +ga- b ta xH1L N
-
-
ab
b
-
b
a
ta t b M $ Iga- b xH1L
xH1L M
-
-
ab
b
-
and
Cosf =
Jg
2 $ Ig
ta xH2L +ga- b ta xH2L N
-
-
ab
b
b
b
a
ta t b M $ Iga- b xH2L
xH2L M
-
-
ab
-
-
b
and multiplying equation (2.8) respectively by J2 ga -b xH1L
N
-
b
J2 ga -b xH2L
N , we get
-
(2.11)
and
1ê2
Sinq Cosa = gKH1L + w 9KH1L d logI KH1L
* M ë ds K
b dVa
-KHnL Wab xH1L
ds
and
(2.12)
1ê2
=
Sinq Cosf = wIKH1L KH2L - KHnL Wab
H1L
dVa
ds
M
where we have used (2.3).
Defining V =
KH1L
KHnL
and eliminating g and w from the equations
(2.6b), (2.11) and (2.12), we obtain
Chapter-III.nb
102
b
HSinq Cosa - V CosqL IV KH2L - Wab xH2L
(2.13)
= Sinq Cosf 9 dV
ds
a
where the relation
* 2 K2 = 1 +
KH1L
HnL
dVa
M
ds
b dVa
- H1 + V 2 L Wab xH1L
=,
ds
2
KH1L
2
KHnL
has been used in the simplification
We, now, discuss the following particular cases:
3.
PARTICULAR CASES
We shall consider the solution of equation (2.13) in the following
two particular cases:
CASE (I): Let Ita , ta M be orthonormal to the first binormal vectors
-
a
a
IxH2L
, xH2L
M. In particular, let Ita , ta M be along the principal normal vector
-
-
a
a
IxH1L
, xH1L
M.
-
This implies Cosf=0. Hence, we have either
(3.1)
KH1L
KHnL
= tanq Cosa,
or
(3.2)
y
M I dV
M xH2L
KH1L KH2L = Wab Wgy I dV
ds
ds
a
b
and their conjugates, where we have used the relation
M I dV
M
KHnL = WabI dV
ds
ds
a
b
in the later equation.
-
CASE (II): Let the congruence l** be along the normal vector N of
the hypersurface.
We have, then Cosq=1, Sinq=0. Since the curve is non-geodesic,
Chapter-III.nb
103
i.e., V∫0, the equation (2.12) reduces to (3.2).
An special feature of (3.2) is in fact the product of the curvatures (with
respect to the hypersurface) of order one and two have been expressed in
terms of the second fundamental tensor of the hypersurface.
Since the curve is non-geodesic and non-asymptotic, we have the
following proposition from (3.2):
A necessary and sufficient condition that the curvature of order two
(with respect to the hypersurface) of a non-geodesic and non-asymptotic
hypernormal curve of order one to be zero is that the first binormal vector
is conjugate with respect to its tangent vector.
On the basis of above discussions, we shall now define and study
some theorems on special, union and hyper-asymptotic curves of a Hermitian hypersurface.
4.
UNION AND SPECIAL CURVES OF A
HERMITIAN HYPERSURFACE
The
first
two
Serret-Frenet's
formulae
for
any
curve
C : zi = zi HsL ; z i = z i HsL of the hypersurface Hnc are given by the equations
-
-
(2.1) and (2.1*).
The components
Iqi ,
q i M and I pa , pa M of the first curvature vectors
-
-
c
of Hnc and Hn+1
are connected by the relations:
(4.1)
qi = pa Bia + Kn* N i ,
(4.1*)
q i = pa Ba-i + K n N i ,
-
-
-
*
-
Chapter-III.nb
104
where
Here,
IN i ,
Bia =
-
-
∑zi
∑ ua
and Ba-i =
∑ zi
∑ ua
.
N i M are the components of the unit normal vector and IKn* , K n M
-
*
is the normal curvature of the hypersurface Hnc in the direction of the curve
C and
(4.2)
i
q =
dzi
ds
, b
du b
ds
-
-
q =
dz i
ds
pa
dua
ds
i
=
-
a
p =
du b
ds
, b
, b
-
dua
ds
+
, b
du b
ds
dz i
ds
+
du b
ds
+
du b
ds
,b
-
-
-
+
-
-
dzi
ds
du b
ds
,b
-
dua
ds
-
,b
du b
ds
-
-
du b
ds
dua
ds
,b
-
,
where
-
Bia , b = Wab N i ; B -i
-
-
a, b
Kn =
a
Wab du
ds
b
du
ds
-- N i ,
= Wab
-
; Kn =
dua
-Wab
ds
-
du b
ds
.
-- are the components of the second fundamental tensor to
and Wab and Wab
the Hermitian hypersurface.
Consider the two congruences of the curve C, given by Il* , l* M and
IL* , L* M, which are such that at each point of Hnc , we have
-
(4.3)
-
-
-
li = ta Bai + L N i ; l i = ta Ba-i + L N i ,
and
(4.3*)
i
a
L =s
Bai
i
-
i
+M N ; L =s
-
a
-
Ba-i
-
+ M N i.
The special and union curves relative to the congruence Ili , l i M in
-
the Riemannian hypersurface have been defined by Tsagas [182] and
Springer [165] respectively.
Chapter-III.nb
105
DEFINITION: SPECIAL CURVE & UNION
CURVE
Let IKl* , K l- M be the special curve relative to the congruence
*
Il* , l* M and the union curve relative to the congruence IL* , L* M in the
Hermitian hypersurface Hnc . Now, this curve of congruence is said to be a
special curve relative to the congruence Il* , l* M, if the vector Il* , l* M lies
in the variety spanned by the vectors
I pa
Bia ,
-
pa
Bai M
-
-
and
Iqi ,
q i M and a
-
curve of the congruence is said to be union curve relative to the congruence IL* , L* M, if the vector IL* , L* M lies in the variety spanned by the
vectors J
dzi
ds
-
,
dz i
ds
N and Iqi , q i M.
-
MATHEMATICAL FORMULATION: Suppose that HG, GL be a
IKl* , K l- M curve relative to the congruence Il* , l* M and a union curve
*
relative to the congruence IL* , L* M, then the above definition mathematically yields:
Ili , l i M = IV , V M I pa Bai , pa Ba-i M + Hw, wL µ
-
-
(4.4)
-
-
-
µ Iqi , q i M
-
and
(4.4*)
IL* , L* M = Hu, uL J
-
dzi
ds
-
,
dz i
ds
N + Hz,
zL Iqi ,
-
q i M,
-
which yield
(4.5)
li = Vpa Bai + w qi and its conjugate,
(4.5*)
L* = u
dzi
ds
z pa and its conjugate.
Using equations (4.1), (4.1*), (4.2), (4.3) and (4.3*), we get
(4.6)
ta = HV + wL pa , N = V Kn* ;
ta = HV + wL pa , N = V K n
-
(4.6*)
-
-
*
Chapter-III.nb
106
and
(4.7)
sa = u
(4.7*)
sa
-
dua
ds
z pa , M = z Kn* ;
-
=
a u du
z
ds
-
*
-
pa , M = z K n .
Defining
2
-
-
2
-
-
-- t a t b ,
T 2 = gab ta t b ; T = gab
-- sa s b ,
S 2 = gab sa s b ; S = gab
2
KH1L
= gab
pa
pb ;
2
K H1L
= gab
--
-
pa
-
pb
and
-- ta t b í T SN,
Cosq = Jgab ta t b ê T S ; gab
-
-
where KH1L is the first curvature vector of the curve C in Hnc . Using (4.6)
and its conjugate, (4.7) and its conjugate, we find
(4.8)
Kn S Cosq = D KH1L ; K n S Cosq = D K H1L .
From the last equation and the equations
-
(4.9)
-
Li L j =
1 = gij Li L j = S 2 + D2 ; 1 = g-ij
2
=S +D
2
and
(4.10)
2
we obtain
(4.11)
9
and
(4.11*)
where
*2
2
2
2
KH1L
= kH1L
+ Kn* 2 ; K H1L = k H1L + K n ,
9
K * n = e KH1LH1 - S 2 L .H1 - S 2 Sin2 qL
1ê2
K
*
2
n = eKH1L I1 - S M .I1 - S Sin qM
2 1ê2
2
kH1L = e KH1L S Cosq ë H1 - S 2 Sin2 qL
2
2
1ê2
1ê2
k H1L = e K H1L S Cosq í I1 - S Sin qM
-
1ê2
1ê2 ,
Chapter-III.nb
107
e = ±1 in order that eCosq be non-negative.
Since IS, SM and Cosq depend upon the congruence Il* , l* M and IL* , L* M,
we have the following theorems from the equations (4.11) and (4.11*):
THEOREM (4.1): If a special curve relative to a fixed congruence
Il* , l * M is a union curve relative to the another fixed congruence
IL* , L* M, then the modulus of the normal and the first curvature with
respect to Hnc at a point of the curve are proportional to its first curvature
with respect to Hnc+1 .
THEOREM (4.2): If the components of the vector fields Il* , l * M
and IL* , L* M tangent to the hypersurface are in the same direction, the
ratio of the two first curvatures is equal to the magnitude of the tangential
components (to the hypersurface) of the Il* , l * M.
PROOF: Since Ita , ta M and Isa , sa M are along the same direction
-
-
Cosq = 1, Sinq= 0,
therefore from the equations (4.11) and (4.11*), we have
kH1L
KH1L
=S;
k H1L
K H1L
= S,
which proves the theorem mentioned above.
5.
HYPER-ASYMPTOTIC CURVE
DEFINITION: HYPER-ASYMPTOTIC CURVE
A curve of the hypersurface is said to be a hyper-asymptotic curve
relative to the congruence IL* , L* M, if the vector IL* , L* M lies in the variety spanned by the vectors
i ,
IhH0L
i M
hH0L
-
and
i ,
IhH2L
i M.
hH2L
-
Chapter-III.nb
108
MATHEMATICAL FORMULATION: A hyper asymptotic curve
of order one, relative to the congruence IL* , L* M in the Hermitian space, is
characterized as follows:
-
(5.1)
-
-
-
-
Li = u hi H0L + zH1L hi H2L ; L i = u h i H0L + zH1L h i H2L .
From the first two Serret-Frenet's formulae (referred to the equa-
tions (2.1) and (2.1*)), we have
(5.2)
d
= -K 2 H1L hi H0L + 8 ds
Hlog KH1L L< qi +
dqi
ds
+KH1L KH2L hi H2L
and
-
(5.2*)
dq i
ds
= -K
2
d
i
i
H1L h H0L + 8 ds Hlog K H1L L< q +
-
-
-
+K H1L K H2L h i H2L .
Another expression for J
dqi
ds
-
,
dq i
ds
N may be obtained by equations (4.1) and
(4.1*) and the relation (Mishra [126]):
(5.3)
d Bi a
ds
= Wab I
du b
ds
M
-
Ni
;
d B i ads
dug
ds
;
= Wab J
--
-
du b
ds
N Ni
-
and
dNi
ds
= -W bg g ba Bai
-
dN i
ds
--
-
-
i
-- g ba B = -W bg
a
The last expression for J
dqi
ds
-
,
dq i
ds
dug
ds
.
N and the equations (5.1), (5.2) and (5.2*)
may be used to eliminate Ihi H2L , h i H2L M with the help of equations (4.1),
-
(4.1*), (4.3) and (4.3*), we have
(5.4)
sa = uH1L
-
(5.4*)
sa
=
-
dua
ds
a
+ yI dpds - Kn* W bg g ba
2
- pa dHlog KH1L ê dsL + KH1L
-
a
uH1L du
ds
-
a
+ yJ
- p dHlog
dug
ds
dua
M,
ds -a
*
dp
dug
-- g ba
K
W
n bg
ds
ds
a
2
K H1L ê dsL + K H1L du
ds
-
N
-
-
Chapter-III.nb
109
and
D = yIWab pa
(5.5)
dua
ds
+
dK*n
ds
+
dKn
ds
- Kn* dHlog KH1L ê dsLM
where
ZH1L
KH1L KH2L
y=
D=
(5.5*)
*
-
-
dua
-- pa
yJWab
ds
- K n dHlog K H1L ê dsLN,
*
where
-
-
y=
Z H1L
K H1L K H2L
.
a
a
a
a
a
a
, xH0L
M, IxH1L
, xH1L
M and IxH2L
, xH2L
M be the unit tangent, unit
Let IxH0L
-
-
-
principal normal and the unit first binormal vectors respectively and
IkH1L , k H1LM, IkH2L , k H2LM be the first and second curvatures of the curve with
respect to the hypersurface.
We obtain from the first two Serret-Frenet's formulae with respect
to the hypersurface Hmc
(5.6)
dpa
ds
dua
+
ds
a
+kH1L kH2L xH2L
dpa
ds
dua
+
ds
a .
+k H1L k H2L xH2L
and
(5.6*)
2
= -kH1L
2
= -k H1L
pa dHlog KH1L L ê ds +
pa dHlog K H1L L ê ds +
-
From the equations (4.10), (5.4), (5.4*), (5.5), (5.5*), (5.6) and (5.6*) and
the definitions
Cosf = J
we deduce
Igab sa
S
du b
ds
M
N,
i Jgab sa
-
--
k
uH1L = IS Cosf, S CosfM
and
S
-
du b
ds
N
y
{
,
Chapter-III.nb
(5.7)
110
AWab pg
du b
ds
dK*n
ds
+
µ Isa - S Cosf
= D . AKn* 2
dua
ds
KH1L
ds
- Kn* dIlog
M
dua
ds
+ pa dI
logkH1L
kH2L
M ë ds +
*
- Kn* W bg g ab
+kH1L kH2L xH2L
b
AWab pg du
ds
-
-
(5.7*)
--
µ Jsa
-
*
dKn
ds
+
*
- K n dJlog
- S Cosf
a
*2
. AK n du
ds
-
=D
N
-
dua
ds
-
+ pa dJ
*
+k H1L k H2L xH2L
log k H1L
k H2L
*
- Kn
ME µ
K H1L
ds
dug
ds
E
NE µ
N í ds +
--
-
dug
-- g ab
W bg
ds
E.
THEOREM (5.1): A hyper-asymptotic curve relative to the congru ence IL* , L* M is characterized by the equations (5.7) and (5.7*).
PROOF: Multiplying equation (5.7) by ga t pt and the equation
-
(5.7*) by ga- -t p t and simplifying, we find
Wab pa
du b
ds
IS Cosf + D
Kn*
kH1L
M=
= AD kH1L dIlogI KH1LH1L MM ë ds - Kn* S Cosf µ
k
µ dIlog I KKH1Ln MM ë dsE ,
*
where we have defined
Cosf =
and
Hgab pa s b L
kH1L S
Chapter-III.nb
111
-
-
du b
-- pa
Wab
ds
JS Cosf + D
= AD k H1L dJlogJ
µ dJlog J
k H1L
K H1L
*
K n
K H1L
*
Kn
k H1L
N=
NN í ds - K n S Cosf µ
*
NN í dsE ,
where we have defined
Jg -- pa s b N
-
-
Cosf =
6.
ab
k H1L S
.
HYPER-ASYMPTOTIC AND SPECIAL CURVES
Let a hyper-asymptotic curve with respect to the congruence
IL* , L* M be a special curve relative to Il* , l* M. this implies
f = q.
Denoting
X = kH1L ê KH1L ; X = k H1L ê K H1L ,
Y = Kn* ê KH1L ; Y = K n ê K H1L ,
*
-
-
a
a
writing pa = kH1L xH1L
; pa = k H1L xH1L
and differentiating the well known
identities
2
2
X 2 + Y 2 = 1 ; X + Y = 1,
we have
(6.1)
D
dX
ds
(6.1*)
D
dX
ds
and
- S Cosq
dY
ds
a I du M µ
= Wab xH1L
ds
- S Cosq
dY
ds
du
-- x a J
= Wab
H1L ds N µ
b
µ HX S Cosq + D Y L ;
µ IX S Cosq + D Y M
-
b
Chapter-III.nb
112
(6.2)
X
(6.2*)
X
dX
ds
+Y
dY
ds
+Y
dY
ds
____
dX
ds
=0;
____
=0.
We shall discuss the solutions of the above equations in the following cases:
a , xa M is not conjugate with respect to
CASE (I): The vector IxH1L
H1L
-
J du
,
ds
a
-
dua
ds
a
N, i.e.,IWab xH1L
A =
are non-singular.
du b
ds
a
∫ 0, Wab xH1L
du b
∫
ds
—
0M and the matrices
i D -S Cosf y
i D -S Cosf y
; B =
—
Y
k X
{
Y
X
k
{
Equations (6.1), (6.1*), (6.2) and (6.2*) yield
b
a du
IWab xH1L
ds
dX
ds
=
dY
ds
a
= -IWab xH1L
MY ;
dX
ds
=
du b
-- x a
JWab
H1L ds
-
-
NY
and
___
dY
ds
-
-- x a
= -JWab
H1L
du b
ds
-
du b
ds
M X;
NX .
The quantities
X = kH1L ê KH1L ; X = k H1L ê K H1L
and
Y = Kn* ê KH1L ; Y = K n ê K H1L
*
are obtained by the solution of the above foregoing simultaneous equations.
CASE (II): Let
a ,
HxH1L
a L
xH1L
be conjugate with respect to J
dua
ds
-
,
dua
ds
N
and the matrices A and B are non-singular.
Equations (6.1), (6.1*), (6.2) and (6.2*) have unique solution given
by the following:
THEOREM (6.1): If a special curve relative to the congruence
Chapter-III.nb
113
Il* , l * M is hyper-asymptotic curve relative to the vector IL* , L* M and
conditions mentioned above in the cases I and II are not satisfied, then the
ratio J
kH1L
KH1L
,
k H1L
K H1L
N and J
*
KH1L
KHnL
N are the same at each point of the curve.
*
K H1L
,
K HnL
CASE (III): The conditions mentioned in case I and case II are not
satisfied, i.e., the matrices A and B are non-singular matrices.
b
a du
JWab xH1L
ds
-
-
;
b
a du
Wab xH1L
ds
--
N may or may not be zero. We have then
*
D Kn* + kH1L S Cosq = 0 ; D K n + k H1L S Cosq = 0,
which in view of (4.9) and (4.10) yields the following:
THEOREM (6.2): If a HKl , K l L curve is hyper asymptotic relative
to IL* , L* M and the conditions mentioned in the cases I and II are not
satisfied, then
(6.3)
kH1L =
KH1L H1-S 2 L
H1-S 2 Sin2 qL
K H1L J1-S N
2 1ê2
1ê2
1ê2
; k H1L =
J1-S Sin2 qN
2
1ê2
and
(6.3*)
Kn* = -
KH1L S Cosq
H1-S 2
2
Sin qL
ê12
*
; Kn = -
K H1L S Cosq
J1-S Sin2 qN
2
ê12
Comparing (6.3), (6.3*), (4.11) and (4.11*), we get the following theorem:
THEOREM (6.3): If two special curves HG , G L and HG * , G * L rela-
tive to the congruences Il* , l * M and IL* , L* M respectively and the conditions expressed in the cases I and II are not satisfied, then the modulus of
the normal curvature in the direction of HG , G L is equal to the first curva-
ture of HG * , G * L and the first curvature of the curve HG , G L is equal to the
modulus of the normal curvature in the direction of HG * , G * L.
Chapter-III.nb
7.
114
HYPER-DARBOUX LINES IN A TACHIBANA
HYPERSURFACE
Let us consider a curve C : zi = zi HsL ; z i = z i HsL (not a geodesic of
-
-
the enveloping space) of Tnc . The components
and
i ,
IxH2L
i ,
IxH0L
i M
xH0L
-
ªJ
dzi
ds
-
,
dz i
ds
i , xi M
N,IxH1L
H1L
-
i M of the unit tangent, unit principal normal and the unit first
xH2L
-
binormal vectors define an orthogonal system of unit vectors at every point
of the curve.
Assuming that d/ds is the usual covariant differential along C, we
have the first two Serret-Frenet's formulae for a curve in Tnc , which are
given by:
(7.1)
i
dxH0L
ds
(7.1*)
i
dxH0L
ds
-
i ;
= KH1L xH0L
-
i
= KH1L xH0L
and
(7.2)
i
dxH1L
ds
(7.2*)
i
dxH1L
ds
-
i
i
= -KH1L xH0L
+ KH2L xH2L
;
-
-
i
i
= -KH1L xH0L
+ KH2L xH2L
,
where the scalars
HKH1L , K H1LL ª J R1H1L ,
1
RH1L
N
HKH2L , K H2LL ª J R1H2L ,
1
RH2L
N
and
are the first and second curvatures of the curve in the embedding space.
Consider a congruence of a curve given by the vector field
Chapter-III.nb
115
c
l** = Ili , l i M, such that through each point of Tn+1
, there passes exactly
-
one curve of congruence. At the point of hypersurface, we have
(7.3)
li = ta Bai + C N i ;
-
(7.3*)
li
=
-
ta
-
-
Ba-i + C N i ,
-
where ta and C are the parameters and ta and C are their complex conjugates respectively.
The curve C is said to be a hyper-D line of the hypersurface
spanned by the vectors
i +R
IRH1L xH1L
H2L
i ,
IxH0L
dRH1L
ds
contain the vector Ili , l i M.
-
i M and
xH0L
-
-
i R xi +R
xH2L
H1L H1L
H2L
i M,
xH2L
-
dRH1L
ds
From (7.1) and (7.1*), we have
(7.4)
i
d2 xH0L
dKH1L
ds
2
i
= -KH1L
xH0L
+
ds2
i
i
xH1L
+ KH1L KH2L xH2L
and
-
(7.4*)
i
d2 xH0L
=
ds2
2
i
-K H1L xH0L
____
dKH1L
ds
+
-
i
xH1L
-
i .
+ K H1L K H2L xH2L
These equations give
(7.5)
i
gi -j IRH1L xH1L
+ RH2L
dRH1L
ds
and
(7.5*)
i
g-i -j JRH1L xH1L
(7.6)
i
+ RH2L
li = uARH1L xH1L
-
-
dR
+ RH2L dsH1L
i
xH2L
MJ
i
d2 xH0L
ds2
N = 0;
2 i
i N i d xH0L
xH2L
ds2
-
-
k
For the hyper-D line of the hypersurface, we get
dRH1L
ds
y
= 0.
{
i
i
xH2L
E + V xH0L
;
and
-
(7.6*)
li
=
i
uARH1L xH1L
-
-
-
dR
+ RH2L dsH1L
We, now, have the following:
i E+V
xH2L
-
-
i .
xH0L
Chapter-III.nb
116
-
(7.7)
lHhL = 2 gi -j l xHhL
i
j
and
-
(7.7*)
lHhL = 2 g i j l i xHhL .
-
j
i , 2 g- J
Multiplying the equation (7.6) by 2 gi -j xH0L
i j
i
d2 xH0L
ds2
N and using the
equations (7.5) and (7.7), we find
i d2 x j y
i d2 x j y
i .
(7.8)
2 gi -j ds2H0L li = 2 gi -j ds2H0L lH0L xH0L
k
{
k
{
From (7.4) and (7.8), we have
-
(7.9*)
I
dKH1L
ds
-
M lH1L + KH1L KH2L lH1L = 0.
Similarly, with the help of equations (7.5*), (7.6*) and (7.7*), we
have
(7.9**)
J
-
dKH1L
ds
N lH1L + K H1L K H2L lH1L = 0.
The equations (7.9*) and (7.9**) represent the hyper-D lines of Tnc relative
to l** .
We, now, have the following:
THEOREM (7.1): If the congruence l** is along the principal
normal, i.e., (along the vector 9xH1Li , xH1Li =) of a curve, then the necessary
-
and sufficient condition that it be a hypersurface is that it be a curve of
constant first curvature.
PROOF: The proof follows immediately from the equations (7.7),
i
i
(7.7*) and (7.9**) and the fact that the vector IxHhL
, xHhL
M constitute an
-
orthogonal system.
In view of theorem (7.1), a hyper-D line satisfies the following:
THEOREM (7.2): If the congruence l** lies along the first binor-
Chapter-III.nb
117
mal then the necessary and sufficient that it be a hyper-D line is that it be a
curve of second curvature.
PROOF: The proof follows from the equations (7.9*), (7.9**) and
c .
the fact that the curve is not a geodesic of Tn+1
We are familiar with the well known relations:
(7.10)
= pa Bai + Wab I du
M I du
M Ni
ds
ds
i
dxH0L
ds
a
b
and
-
(7.10*)
i
dxH0L
ds
=
-
pa
-
Ba-i
+ Wab J
--
-
dua
ds
NJ
-
du b
ds
N Ni.
-
Writing the covariant differentials of equation (7.10) and using
(Mishra [67], page 155):
N,i g = W bg g bg Bai + fg N i ,
we get
(7.11)
i
d2 xH0L
2
ds
= 9 dpds - Wmb
a
µ Bai A3 Wab
du b
ds
dua
+Wab I ds
dum
ds
du b
ds
Wgn g ga
pa + Wab,g
M I du
M fg I
ds
b
dun
ds
=µ
dua du b dug
ds
ds
ds
dun
i
ME.N .
ds
+
Similarly, writing the covariant differentials of (7.10*) and using (Mishra
[67], page 155):
--
-
-
-
-- g bg B -i + f - N i ,
N,ig- = W bg
g
a
we get the conjugate of the equation (7.11) as follows:
-
(7.11*)
i
d2 xH0L
ds2
=9
-
µ Ba-i
-
dpa
ds
-
dum
-- Wmb
ds
du b
-A3 Wab
ds
-
a
+Wab J du
ds
-
--
-
a
p
NJ
-
du b
ds
--
-- g ga
Wgn
-
dua
-- + Wab
,g ds
-
du b
ds
N fg J
-
-
dun
ds
-
dun
ds
-
=µ
-
du b
ds
dug
ds
NE.N i .
+
-
Substituting from (7.11) in (7.8) and using the fact that
i ,
I xH0L
i M is orthogxH0L
-
Chapter-III.nb
118
onal to IN i , N i M, we get
-
I dpds M ta - W mb
a
n
dum du b
g du
W
t
+
gn
ds
ds
ds
b
dua du b dug
a
p
+
W
+CA3 Wab du
ab,g
ds
ds
ds
ds
dua du b dug
+Wab ds ds ds fg E +
2
a
dpa du b
du b
M
g
I
+lH0LAIWab du
ab ds M ds E
ds
ds
(7.12)
+
= 0.
Similarly, the conjugate of the equation (7.12) is given by
J
(7.12*)
-
dpa
ds
m
N ta - Wmb du
ds
-
-
-
-
-
du b
-+CA3 Wab
ds
p
a
n
Wgn tg du
ds
a
+lH0LAJWab du
ds
-
-
-
-
dua
-- + Wab
,g ds
-
du b
ds
du b
ds
--
+
--
-
-
dua
-+Wab
ds
-
-
du b
ds
--
dug
ds
N
2
du b
ds
fg- E +
a
- gabJ dpds
-
--
N
-
dug
ds
-
du b
ds
+
E = 0.
Now, we have the first two Serret-Frenet's formulae for Tnc given by
pa =
(7.13)
d
ds
a ,
I du
M = KH1L zH1L
ds
a
a
= -KH1L I du
M + KH2L zH2L
ds
a
dzH1L
ds
a
with its conjugate.
a
a
zH1L
and zH2L
are the components of the principal and binormal vec-
tors with respect to Tnc . These equations give
I dpds M = -K 2 H1LI du
M+
ds
a
a
a
.
+KH1L KH2L zH2L
dKH1L
ds
a
zH1L
+
Substituting the values of I dpds M in (7.12) and using the fact that
a
lH0L = 2 gij- l
i
we have
-
j
xH0L
-
=
b
2 gab- ta du
ds
def
= tH0L ,
Chapter-III.nb
119
dKH1L
(7.14)
ds
dua
ds
tH1L + KH1L KH2L tH2L - Wab
du b
ds
µ
µ Wgn tg I du
M + 3 CWab pa I du
M+
ds
ds
n
b
+CWab,g I du
M I du
M H du
L+
ds
ds
ds
a
b
g
M I du
M fg I du
M+
+C Wab I du
ds
ds
ds
a
+tH0L IWab
b
dua
ds
n
M = 0,
2
du b
ds
i and z i
where tH1L and tH2L are the projections of ta in the direction of zH1L
H2L
respectively.
In the same way, we may obtain the conjugate of (7.14) as
dKH1L t H1L
(7.14*)
ds
+ K H1L K H2L t H2L
n
µ Wgn tg J du
ds
-
-
--
dua
-- - J
+CWab
,g ds
-
dua
-- J
+C Wab
ds
-
+ t H0L
NJ
-
-
du b
ds
b
N + 3 CWab pa J du
ds
-
--
NJ
a
JWab du
ds
--
-
dua
-- Wab
ds
-
du b
ds
-
du b
ds
-
du b
ds
NJ
-
dug
ds
N fg- J
N+
N = 0.
2
-
dun
ds
-
µ
N+
N+
The equations (7.14) and (7.14*) represent the hyper-D lines of the
hypersurface.
The above equations have been represented un terms of the second
fundamental tensors and the curvatures of the curve with respect to the
hypersurface.
Chapter-III.nb
120
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