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GENERALIZED CIRCLES AND THEIR CONFORMAL MAPPING IN A SUBSPACE OF A WEYL SPACE
2005.25B(2):331-339
GENERALIZED CIRCLES AND THEIR CONFORMAL MAPPING IN A SUBSPACE OF A WEYL SPACE 1 Giiler Giirpuuu: A rsan
Gulr;in Qivi Ysldsrim
Department of Mathematics, Faculty of Science and Letters, Istanbul Technical University 34469, Maslak-Istanbul, Turkey E-mail: [email protected]; E-mail: [email protected]
Abstract The authors give the necessary and sufficient conditions for a generalized circle in a Weyl hypersurface to be generalized circle in the enveloping Weyl space. They then obtain the neccessary and sufficient conditions under which a generalized concircular transformation of one Weyl space onto another induces a generalized transformation on its subspaces. Finally, it is shown that any totally geodesic or totally umbilical hypersurface of a generalized concircularly flat Weyl space is also generalized con circularly flat. Key words Weyl space, generalized circle, generalized concircular transformation, totally umbilical space, generalized concircular flat space 2000 MR Subject Classification
1
53A40, 53A30
Introduction
Circles in Riemannian spaces and their conformal transformations have been studied in [1],[2],[3]. The theory of Riemannian subspaces in the concircular geometry has been given in [4] by Yano, K. In [5], Li, Zl. have obtained a characteristic of Riemannian spaces admitting quasiconcircular transformation. In [6], Nomizu, K. and Yano, K. have studied circles and spheres in Riemannian spaces and have given the necessary and sufficient condition for every circle in a submanifold M" of a Riemannian manifold M'" to be a circle in M'": In [7], generalized circles in a Weyl space have been defined and conformal mappings preserving generalized circles (the generalized concircular mappings) have been studied. In this work, generalized circles in Weyl hypersurfaces and generalized concircular mappings of Weyl subspaces have been studied. An n-dimensional differentiable manifold W n is said to be a Weyl space if it has a conformal metric tensor 9 and a symmetric connection V' satisfying the compatibility condition given by the equation
V'"( 90.(3
-
2 T"(
90.(3
= 0,
(1.1)
where T"( denotes a covariant vector field. Under the renormalization
(1.2) 1 Received
January 15, 2003
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Vol.25 Ser.B
of the metric tensor g, T is transformed by the law
i; = T"( + 0, In >..
(1.3)
where>" is a function defined on W n . An object A defined on Wn(g, T) is called a satellite of weight {p} of the tensor ga(3, if it admits a transformation of the form A= >..PA (1.4) under the renormalization of the metric tensor ga(3 [8,9J. The prolonged covariant derivative of a satellite A is defined by
"9, A = \7, A -
p T, A .
(1.5)
We note that the prolonged covariant derivative preserves the weight. Writing (1.1) in local coordinates and expanding it we find that
o,ga(3 - g/-,(3 r~"( where
r~"(
-
ga/-, r~,
-
2 T, ga(3
=0
(1.6)
are the connection coefficients of the form
r~, =
{;')' } - (8$ T"(
+ 8~ T(3 _g'ia g(3,T8).
(1.7)
Let Wm(gij,Tk) be a subspace with coordinates yi (i = 1,2,···,m) of the Weyl space W n(ga(3, T"() with coordinates x a (a = 1,2,···, m, m + 1,···, n). Suppose that the metrics of W m (gij, Tk) and W n (ga(3, T,) are elliptic and that they are given, respectively, by gij dyi dyj and which are connected by the relations
gij=ga(3XiX~ (i,j=I,2,···,mi a,,8=1,2,···,n),
(1.8)
where xi denotes the covariant derivative of x a with respect to yi . The prolonged covariant derivatives of the satellite A, relative to W m and W n , are related by (1.9) Denote by n a (0" = 1,2,···, n - m) , the contravariant components of the n - m linear a
independent unit vector fields
ti
a
in W n normal to W m
.
The moving frame {x~, f;.a} on W m, reciprocal to the moving frame {xi, n a}, is defined by the relations [9]
a
(JL = 1,2,···, n - m) .
(1.10)
Remembering that the weight of xi is {O}, the prolonged covariant derivative of (1.10)4 with respect to yk is obtained as [10] (1.11) where Wik is the second fundamental form of Wm(gij, T k). It can be easily seen that Wik is a satellite of gij of weight {I}.
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No.2
Generalized Circles in Weyl Hypersurfaces
Let Wn-1(gij,Tk) be a hypersurface, with coordinates yi (i = 1,2,···,n -1) of a Weyl space W n (ga(3, T,) with coordinates x" (a = 1, 2, ... , n). The relation between the metrics of W n - 1 and is
w,
gij=ga(3X?X%.
(2.1)
(i,j=1,2,···,n- 1 i a,{3=1,2,···,n)
Consider the moving frame {x~,na} in W n- 1 reciprocal to the moving frame {x?,n a}. The relations (1.10) and (1.11) become (2.2) ~ V
jXia
= r7v jX ia = Wij n a
(2.3)
.
Let n a be the contravariant components of the vector field in W n normal to W n is normalized by the condition
1,
which (2.4)
Let vi be the contravariant components of the vector field v in W n by the condition gij vi v j = 1 .
1,
which is normalized
On the other hand, if the components of v relative to W n is denoted by VA, we have (2.5) The absolute derivative[ll] of the vector field VA relative to W n in the direction of v k, according to (2.3) and (2.5), is (2.6) Taking the absolute derivative of the equation (2.6) in the direction of v Z and using (2.3), we have
VZt'I(Vkt'k VA) = vi t'z [( Wik vi v k) n A + x; Vk (t'k vi)] = vi t'Z ( Wik Vi v k) n A + (Wik vi v k) v Zt'Z n A +V l (Wi! n A) Vk (t'kvi) + x; (v Zt'zv k) (t'kvi)
(2.7)
+ x; Vk v Z(t'l t'kvi)
.
In [7], generalized circles in a Weyl space are defined as follows:
Let C be a smooth curve belonging to W n and let v be the tangent vector to C at the point P normalized by the condition ga(3 va v(3 = 1 . C is called a generalized circle if there exist a vector field u, normalized by the condition g(u, v) = 1, along C and a positive prolonged covariant scalar function K, of weight {-I} such that (2.8) From (2.8) it follows that v Zt'z (v k t' kVA ) + VA gJll' v k (t' kVJl) vi (t'IVl') = 0 .
(2.9)
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We now prove Theorem 2.1A generalized circle with a non-asymptotic direction v, belonging to the hypersurface W n - 1 will be a generalized circle relative to the enveloping space W n if and only if the conditions
are satisfied, where
Kn
is the normal curvature of W n -
1
in the direction of v, which is defined
by Kn
Wik V'i Vk
=
Proof By using (2.4), (2.6) and (2.7) we find Vi Vi (v k VkV.\) + V.\ g"v vk(VkV") vi (VIVV) = x; [vi Vi(V k Vk vi) + vi gjt Vk(Vkvj) Vi(V1v t)]
(2.10)
+V.\(Wjk v j vk)(Wtl v t vi) + vi VI(Wik vi v k) n.\ +( Wik Vi v k) vi (Vi n.\) + vi Wil n.\ v k (V kvi)
Suppose now that the generalized circles of W n - 1 are also generalized circles of W n . Then the left hand side of (2.10) and the brackets on the right side both vanish. So, we have
+(WikViVk)Vi(VI n.\) The absolute derivative of
n.\
+ vlwiln.\vk(VkVi)
=
o.
(2.11)
in the direction of vi is given by (2.12)
Substitution of (2.5) and (2.12) in (2.11) gives x; [vi(Wjkvjvk)(Wtlvtvl) - (WikVivk)vmWmzgii] +n.\ [vIVi(WikVivk)
+ vIWilVk(VkVi)] = 0
(2.13)
or, equivalently, (2.14) Equating the tangential and the normal components to zero and remembering
Kn
=I- 0 , we
get (2.15) Conversely, if a curve in W n - 1 is a generalized circle and the conditions (2.15) are satisfied, then such a curve is also a generalized circle relative to W n . Remark 2.1 If a generalized circle belonging to W n - 1 is also an asymptotic line, then instead of the two conditions (2.15) we obtain the single condition
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Generalized Concircular Transformation of a Weyl Subspace
In [7], generalized concircular transformation of a Weyl space upon another Weyl space is defined. In this section, we will consider the generalized concircular transformation of a Weyl subspace.
Let T be a conformal transformation of the Weyl space Wn(g, T) into the Weyl space Wn(.ij, T). Then, at corresponding points of these spaces we can make[9,12] 9 = g.
(3.1)
P=T-T
(3.2)
The covariant vector field P defined by
is called the covector field of the conformal transformation. It can be shown that P has zero weight. Let \7 and '\7 be the Weyl connections of Wn(g, T) and Wn(.ij, T), and the connection coefficients be denoted by f3"Y and f3"Y, respectively. By using (1.7), (3.1) and (3.2), the relation between I' and
t
is obtained as
(3.3) After some calculations, in view of (3.3), the mixed curvature tensor becornesl/l
where P a(3 =
\7(3
e: - r:
P(3
+ ~ u" PI' PI-' ga(3'
(3.5)
The necessary and sufficient condition for the conformal transformation T
Wn(g, T)
---->
Wn(.il, T) to be generalized concircular is that[7] (3.6)
where ¢ is a scalar smooth function of weight {-2} defined on Wn(g, T). Let T be a generalized concircular transformation of W n into Wn and let W m be a subspace of Wn(g, T). In general, T does not induce a transformation on W m of the same kind. In this connection, we have Theorem 3.1 A generalized concircular transformation of a Weyl space induces a con-
circular transformation on a subspace if and only if
(3.7) where
QI-'jk
_
-
D I-' v k xj
-
-
1
m
gjk
9
il
(D 1-') v I Xi .
Proof Let us consider the subspace W m immersed in the Weyl space W n . conformal transformation T of W n into w, be generalized concircular, i.e., let
(3.8) Let the
(3.9)
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We first seek the condition under which W m is generalized concircularly transformed into any other Weyl space. The expression
(3.10) where (3.11) can be written in the form
Pjk = P/-,v X /-' Xkv j
-
1 ",(3 P P . /-' 1 hi 2 9 '" (3 gjk + P/-, "ilk X j + 2 9
Ph PI gjk
(3.12)
.
On the other hand, remembering that the reciprocal fundamental tensors
g",(3
and
gab
of
W n and W m respectively are related by g",(3
=
gab
X~ X~
n
+
~
n'" n(3,
/,=m+l /'
(a, b = 1,2,"', m; n, f3 = 1,2,"', m, m
+ 1""
/'
n)
and by using (3.11), (3.12) can be put into the form (3.13) Substitution of (3.9) in (3.13) gives (3.14) Suppose now that the induced conformal transformation of W m is also generalized concircular, i.e., (3.15) where 'P is a scalar function defined on W m . Substitution (3.15) in (3.14) and transvection by
gjk
yield (3.16)
By virtue of (3.15) and (3.16), (3.14) becomes
where
/-' - v k xj Q jk~ _.r,
-
1
9
i/.r,
/-'
v I xi gjk .
m Conversely, if the equation (3.7) is satisfied, then it is clear that (3.14) reduces to (3.17)
Arsan & Yildmrn: GENERALIZED CIRCLES AND THEIR CONFORMAL MAPPING
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from which it follows that
where
So the proof is completed.
Theorem 3.2 A generalized concircular transformation of a Weyl space induces a generalized concircular transformation on its totally umbilical subspaces. Proof If W m is totally umbilical, we have (3.18)
Substituting (3.18) in (3.14), we get Pjk = 'P gjk ,
where
This implies that the Weyl subspace is a generalized concircular one.
4 Some Special Hypersurfaces of Generalized Concircularly Flat Weyl Spaces The generalized concircular curvature tensors of W n and W n -
1
are, respectively, given by
[7] (4.1)
and ZJkh = Rjkh -
(n _ l;(n _ 2)
(gjk
<5~ -
where Rand r are respective scalar curvatures of W n and W n The equations of Gauss for W n - 1 in W n are[13]
gjh
<5D
(4.2)
1.
(4.3)
or equivalently,
(4.4)
x;
where x~ = gti g)..(3 and gtiRhh = Rijkh. Contracting (4.1) with xi xj X k xl:, we have
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By substituting (4.5) in (4.4), we obtain
u:jkh --
ZA t /-' v W /-,VW x A x j xk xh
+ g ti ( Wik Wjh
- Wih Wjk
)
+ n(nR_
s:t.1) (gjk Uh
(4. 6)
s:t) .
gjh Uk
By contraction on the indices t and h in (4.6), we find that
R jk --
A /-' vAw Z/-,VW X j Xk A
where A>: = x~ x'h. Transvecting (4.7) by
r -_
g
ab
R ab -_
gjk,
+ g hi(Wik Wjh
- Wih Wjk
)
+ (n-2)R n(n _ 1) gjk
(4.7)
we obtain
ZA ab /-' v AW /-,VW g x a Xb A
+ g ab g hi(Wib Wah
-
Wih Wab
)
+ (n -
n
2) R
.
(4.8)
The equations (4.2), (4.6) and (4.8) give
1 _ 2) (gjk s:t (n _ l)(n Uh
-
s:t)
gjh Uk g
ab
g
pl (
)
WZb W a p - WZ p Wab .
(4 . 9)
We know to deduce the following results: Theorem 4.1 A totally umbilical hypersurface of a generalized concircularly flat Weyl space is also generalized concircularly flat. For a totaly umbilical hypersurface W n - 1 , we have 1
Wij
= (n _ 1) M %
In this case, the equation (4.9) becomes
(4.10) from which it follows that, if the enveloping space W n is generalized concircularly flat 0) , then
(Z:vw =
(4.11) stating that W n - 1 is also generalized concircularly flat. Suppose now that Wn - 1 is totally geodesic and that W n is generalized concircularly flat. Then, we have Wij = 0 and Z:vw = 0 . Under these conditions (4.9) gives ZJkh = 0 . So we have proved Theorem 4.2 A totally geodesic hypersurface of a generalized concircularly flat Weyl space is also generalized concircularly flat.
Acknowledgement The authors would like to express their many thanks to professor Abdiilkadir Ozdeger for his valuable suggestions.
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References 1 2 3 4 5
Yano K. Concircular Geometry I, Concircular Transformations. Proc Imp Acad, 1940. 195-200 Yano K. Concircular Geometry II, Integrability Conditions of PV/l- = tPgV/l-' Proc Imp Acad, 1940. 354-360 Yano K. Concircular Geometry III, Theory of Curves. Proc Imp Acad, 1940. 442-458 Yano K. Concircular Geometry IV, Theory of Subspaces. Proc Imp Acad, 1940. 505-511 Li Z L. A Characteristic of Riemannian Spaces Admitting Quasi-Concircular Transformation. Acta Math
Sci, 1991, 11(1): 56-64 6 Nomizu K, Yano K. On Circles and Spheres in Riemannian Geometry. Math Ann, 1974. 163-170 7 Ozdeger A,~entiirk, Z. Generalized Circles in Weyl spaces and their conformal mapping. Publ Math Debrecen, 2002, 60(1/2): 75-87 8 Hlavaty V. Theorie d'immersion d'une W rn dans W n . Ann Soc Polon Math, 1949, 21: 196-206 9 Norden A. Affinely connected spaces (in Russian). Moscow, GRMFML, 1976 10 Ozdeger A. A Chebyshev nets in a subspace of a Weyl space and the generalized Dupin theorem. Webs and Quasigroups, 1995. 97-105 11 Kofoglu N, Ozdeger A. Some special nets of curves in a Weyl hypersurface. Webs and Quasigroups, 2000. 104-113 12 Zlatanov G, Tsareva B. On the Geometry of the nets in the n-dimensional space of Weyl. Journal of Geometry, 1990, 38: 182-197 13 Canfes E 0, Ozdeger A. Some applications of prolonged covariant differentiation in Weyl spaces. Journal of Geometry, 1997, 60: 7-16
GENERALIZED CIRCLES AND THEIR CONFORMAL MAPPING IN A SUBSPACE OF A WEYL SPACE 1 Giiler Giirpuuu: A rsan
Gulr;in Qivi Ysldsrim
Department of Mathematics, Faculty of Science and Letters, Istanbul Technical University 34469, Maslak-Istanbul, Turkey E-mail: [email protected]; E-mail: [email protected]
Abstract The authors give the necessary and sufficient conditions for a generalized circle in a Weyl hypersurface to be generalized circle in the enveloping Weyl space. They then obtain the neccessary and sufficient conditions under which a generalized concircular transformation of one Weyl space onto another induces a generalized transformation on its subspaces. Finally, it is shown that any totally geodesic or totally umbilical hypersurface of a generalized concircularly flat Weyl space is also generalized con circularly flat. Key words Weyl space, generalized circle, generalized concircular transformation, totally umbilical space, generalized concircular flat space 2000 MR Subject Classification
1
53A40, 53A30
Introduction
Circles in Riemannian spaces and their conformal transformations have been studied in [1],[2],[3]. The theory of Riemannian subspaces in the concircular geometry has been given in [4] by Yano, K. In [5], Li, Zl. have obtained a characteristic of Riemannian spaces admitting quasiconcircular transformation. In [6], Nomizu, K. and Yano, K. have studied circles and spheres in Riemannian spaces and have given the necessary and sufficient condition for every circle in a submanifold M" of a Riemannian manifold M'" to be a circle in M'": In [7], generalized circles in a Weyl space have been defined and conformal mappings preserving generalized circles (the generalized concircular mappings) have been studied. In this work, generalized circles in Weyl hypersurfaces and generalized concircular mappings of Weyl subspaces have been studied. An n-dimensional differentiable manifold W n is said to be a Weyl space if it has a conformal metric tensor 9 and a symmetric connection V' satisfying the compatibility condition given by the equation
V'"( 90.(3
-
2 T"(
90.(3
= 0,
(1.1)
where T"( denotes a covariant vector field. Under the renormalization
(1.2) 1 Received
January 15, 2003
332
ACTA MATHEMATICA SCIENTIA
Vol.25 Ser.B
of the metric tensor g, T is transformed by the law
i; = T"( + 0, In >..
(1.3)
where>" is a function defined on W n . An object A defined on Wn(g, T) is called a satellite of weight {p} of the tensor ga(3, if it admits a transformation of the form A= >..PA (1.4) under the renormalization of the metric tensor ga(3 [8,9J. The prolonged covariant derivative of a satellite A is defined by
"9, A = \7, A -
p T, A .
(1.5)
We note that the prolonged covariant derivative preserves the weight. Writing (1.1) in local coordinates and expanding it we find that
o,ga(3 - g/-,(3 r~"( where
r~"(
-
ga/-, r~,
-
2 T, ga(3
=0
(1.6)
are the connection coefficients of the form
r~, =
{;')' } - (8$ T"(
+ 8~ T(3 _g'ia g(3,T8).
(1.7)
Let Wm(gij,Tk) be a subspace with coordinates yi (i = 1,2,···,m) of the Weyl space W n(ga(3, T"() with coordinates x a (a = 1,2,···, m, m + 1,···, n). Suppose that the metrics of W m (gij, Tk) and W n (ga(3, T,) are elliptic and that they are given, respectively, by gij dyi dyj and which are connected by the relations
gij=ga(3XiX~ (i,j=I,2,···,mi a,,8=1,2,···,n),
(1.8)
where xi denotes the covariant derivative of x a with respect to yi . The prolonged covariant derivatives of the satellite A, relative to W m and W n , are related by (1.9) Denote by n a (0" = 1,2,···, n - m) , the contravariant components of the n - m linear a
independent unit vector fields
ti
a
in W n normal to W m
.
The moving frame {x~, f;.a} on W m, reciprocal to the moving frame {xi, n a}, is defined by the relations [9]
a
(JL = 1,2,···, n - m) .
(1.10)
Remembering that the weight of xi is {O}, the prolonged covariant derivative of (1.10)4 with respect to yk is obtained as [10] (1.11) where Wik is the second fundamental form of Wm(gij, T k). It can be easily seen that Wik is a satellite of gij of weight {I}.
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Generalized Circles in Weyl Hypersurfaces
Let Wn-1(gij,Tk) be a hypersurface, with coordinates yi (i = 1,2,···,n -1) of a Weyl space W n (ga(3, T,) with coordinates x" (a = 1, 2, ... , n). The relation between the metrics of W n - 1 and is
w,
gij=ga(3X?X%.
(2.1)
(i,j=1,2,···,n- 1 i a,{3=1,2,···,n)
Consider the moving frame {x~,na} in W n- 1 reciprocal to the moving frame {x?,n a}. The relations (1.10) and (1.11) become (2.2) ~ V
jXia
= r7v jX ia = Wij n a
(2.3)
.
Let n a be the contravariant components of the vector field in W n normal to W n is normalized by the condition
1,
which (2.4)
Let vi be the contravariant components of the vector field v in W n by the condition gij vi v j = 1 .
1,
which is normalized
On the other hand, if the components of v relative to W n is denoted by VA, we have (2.5) The absolute derivative[ll] of the vector field VA relative to W n in the direction of v k, according to (2.3) and (2.5), is (2.6) Taking the absolute derivative of the equation (2.6) in the direction of v Z and using (2.3), we have
VZt'I(Vkt'k VA) = vi t'z [( Wik vi v k) n A + x; Vk (t'k vi)] = vi t'Z ( Wik Vi v k) n A + (Wik vi v k) v Zt'Z n A +V l (Wi! n A) Vk (t'kvi) + x; (v Zt'zv k) (t'kvi)
(2.7)
+ x; Vk v Z(t'l t'kvi)
.
In [7], generalized circles in a Weyl space are defined as follows:
Let C be a smooth curve belonging to W n and let v be the tangent vector to C at the point P normalized by the condition ga(3 va v(3 = 1 . C is called a generalized circle if there exist a vector field u, normalized by the condition g(u, v) = 1, along C and a positive prolonged covariant scalar function K, of weight {-I} such that (2.8) From (2.8) it follows that v Zt'z (v k t' kVA ) + VA gJll' v k (t' kVJl) vi (t'IVl') = 0 .
(2.9)
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We now prove Theorem 2.1A generalized circle with a non-asymptotic direction v, belonging to the hypersurface W n - 1 will be a generalized circle relative to the enveloping space W n if and only if the conditions
are satisfied, where
Kn
is the normal curvature of W n -
1
in the direction of v, which is defined
by Kn
Wik V'i Vk
=
Proof By using (2.4), (2.6) and (2.7) we find Vi Vi (v k VkV.\) + V.\ g"v vk(VkV") vi (VIVV) = x; [vi Vi(V k Vk vi) + vi gjt Vk(Vkvj) Vi(V1v t)]
(2.10)
+V.\(Wjk v j vk)(Wtl v t vi) + vi VI(Wik vi v k) n.\ +( Wik Vi v k) vi (Vi n.\) + vi Wil n.\ v k (V kvi)
Suppose now that the generalized circles of W n - 1 are also generalized circles of W n . Then the left hand side of (2.10) and the brackets on the right side both vanish. So, we have
+(WikViVk)Vi(VI n.\) The absolute derivative of
n.\
+ vlwiln.\vk(VkVi)
=
o.
(2.11)
in the direction of vi is given by (2.12)
Substitution of (2.5) and (2.12) in (2.11) gives x; [vi(Wjkvjvk)(Wtlvtvl) - (WikVivk)vmWmzgii] +n.\ [vIVi(WikVivk)
+ vIWilVk(VkVi)] = 0
(2.13)
or, equivalently, (2.14) Equating the tangential and the normal components to zero and remembering
Kn
=I- 0 , we
get (2.15) Conversely, if a curve in W n - 1 is a generalized circle and the conditions (2.15) are satisfied, then such a curve is also a generalized circle relative to W n . Remark 2.1 If a generalized circle belonging to W n - 1 is also an asymptotic line, then instead of the two conditions (2.15) we obtain the single condition
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Generalized Concircular Transformation of a Weyl Subspace
In [7], generalized concircular transformation of a Weyl space upon another Weyl space is defined. In this section, we will consider the generalized concircular transformation of a Weyl subspace.
Let T be a conformal transformation of the Weyl space Wn(g, T) into the Weyl space Wn(.ij, T). Then, at corresponding points of these spaces we can make[9,12] 9 = g.
(3.1)
P=T-T
(3.2)
The covariant vector field P defined by
is called the covector field of the conformal transformation. It can be shown that P has zero weight. Let \7 and '\7 be the Weyl connections of Wn(g, T) and Wn(.ij, T), and the connection coefficients be denoted by f3"Y and f3"Y, respectively. By using (1.7), (3.1) and (3.2), the relation between I' and
t
is obtained as
(3.3) After some calculations, in view of (3.3), the mixed curvature tensor becornesl/l
where P a(3 =
\7(3
e: - r:
P(3
+ ~ u" PI' PI-' ga(3'
(3.5)
The necessary and sufficient condition for the conformal transformation T
Wn(g, T)
---->
Wn(.il, T) to be generalized concircular is that[7] (3.6)
where ¢ is a scalar smooth function of weight {-2} defined on Wn(g, T). Let T be a generalized concircular transformation of W n into Wn and let W m be a subspace of Wn(g, T). In general, T does not induce a transformation on W m of the same kind. In this connection, we have Theorem 3.1 A generalized concircular transformation of a Weyl space induces a con-
circular transformation on a subspace if and only if
(3.7) where
QI-'jk
_
-
D I-' v k xj
-
-
1
m
gjk
9
il
(D 1-') v I Xi .
Proof Let us consider the subspace W m immersed in the Weyl space W n . conformal transformation T of W n into w, be generalized concircular, i.e., let
(3.8) Let the
(3.9)
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We first seek the condition under which W m is generalized concircularly transformed into any other Weyl space. The expression
(3.10) where (3.11) can be written in the form
Pjk = P/-,v X /-' Xkv j
-
1 ",(3 P P . /-' 1 hi 2 9 '" (3 gjk + P/-, "ilk X j + 2 9
Ph PI gjk
(3.12)
.
On the other hand, remembering that the reciprocal fundamental tensors
g",(3
and
gab
of
W n and W m respectively are related by g",(3
=
gab
X~ X~
n
+
~
n'" n(3,
/,=m+l /'
(a, b = 1,2,"', m; n, f3 = 1,2,"', m, m
+ 1""
/'
n)
and by using (3.11), (3.12) can be put into the form (3.13) Substitution of (3.9) in (3.13) gives (3.14) Suppose now that the induced conformal transformation of W m is also generalized concircular, i.e., (3.15) where 'P is a scalar function defined on W m . Substitution (3.15) in (3.14) and transvection by
gjk
yield (3.16)
By virtue of (3.15) and (3.16), (3.14) becomes
where
/-' - v k xj Q jk~ _.r,
-
1
9
i/.r,
/-'
v I xi gjk .
m Conversely, if the equation (3.7) is satisfied, then it is clear that (3.14) reduces to (3.17)
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from which it follows that
where
So the proof is completed.
Theorem 3.2 A generalized concircular transformation of a Weyl space induces a generalized concircular transformation on its totally umbilical subspaces. Proof If W m is totally umbilical, we have (3.18)
Substituting (3.18) in (3.14), we get Pjk = 'P gjk ,
where
This implies that the Weyl subspace is a generalized concircular one.
4 Some Special Hypersurfaces of Generalized Concircularly Flat Weyl Spaces The generalized concircular curvature tensors of W n and W n -
1
are, respectively, given by
[7] (4.1)
and ZJkh = Rjkh -
(n _ l;(n _ 2)
(gjk
<5~ -
where Rand r are respective scalar curvatures of W n and W n The equations of Gauss for W n - 1 in W n are[13]
gjh
<5D
(4.2)
1.
(4.3)
or equivalently,
(4.4)
x;
where x~ = gti g)..(3 and gtiRhh = Rijkh. Contracting (4.1) with xi xj X k xl:, we have
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By substituting (4.5) in (4.4), we obtain
u:jkh --
ZA t /-' v W /-,VW x A x j xk xh
+ g ti ( Wik Wjh
- Wih Wjk
)
+ n(nR_
s:t.1) (gjk Uh
(4. 6)
s:t) .
gjh Uk
By contraction on the indices t and h in (4.6), we find that
R jk --
A /-' vAw Z/-,VW X j Xk A
where A>: = x~ x'h. Transvecting (4.7) by
r -_
g
ab
R ab -_
gjk,
+ g hi(Wik Wjh
- Wih Wjk
)
+ (n-2)R n(n _ 1) gjk
(4.7)
we obtain
ZA ab /-' v AW /-,VW g x a Xb A
+ g ab g hi(Wib Wah
-
Wih Wab
)
+ (n -
n
2) R
.
(4.8)
The equations (4.2), (4.6) and (4.8) give
1 _ 2) (gjk s:t (n _ l)(n Uh
-
s:t)
gjh Uk g
ab
g
pl (
)
WZb W a p - WZ p Wab .
(4 . 9)
We know to deduce the following results: Theorem 4.1 A totally umbilical hypersurface of a generalized concircularly flat Weyl space is also generalized concircularly flat. For a totaly umbilical hypersurface W n - 1 , we have 1
Wij
= (n _ 1) M %
In this case, the equation (4.9) becomes
(4.10) from which it follows that, if the enveloping space W n is generalized concircularly flat 0) , then
(Z:vw =
(4.11) stating that W n - 1 is also generalized concircularly flat. Suppose now that Wn - 1 is totally geodesic and that W n is generalized concircularly flat. Then, we have Wij = 0 and Z:vw = 0 . Under these conditions (4.9) gives ZJkh = 0 . So we have proved Theorem 4.2 A totally geodesic hypersurface of a generalized concircularly flat Weyl space is also generalized concircularly flat.
Acknowledgement The authors would like to express their many thanks to professor Abdiilkadir Ozdeger for his valuable suggestions.
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References 1 2 3 4 5
Yano K. Concircular Geometry I, Concircular Transformations. Proc Imp Acad, 1940. 195-200 Yano K. Concircular Geometry II, Integrability Conditions of PV/l- = tPgV/l-' Proc Imp Acad, 1940. 354-360 Yano K. Concircular Geometry III, Theory of Curves. Proc Imp Acad, 1940. 442-458 Yano K. Concircular Geometry IV, Theory of Subspaces. Proc Imp Acad, 1940. 505-511 Li Z L. A Characteristic of Riemannian Spaces Admitting Quasi-Concircular Transformation. Acta Math
Sci, 1991, 11(1): 56-64 6 Nomizu K, Yano K. On Circles and Spheres in Riemannian Geometry. Math Ann, 1974. 163-170 7 Ozdeger A,~entiirk, Z. Generalized Circles in Weyl spaces and their conformal mapping. Publ Math Debrecen, 2002, 60(1/2): 75-87 8 Hlavaty V. Theorie d'immersion d'une W rn dans W n . Ann Soc Polon Math, 1949, 21: 196-206 9 Norden A. Affinely connected spaces (in Russian). Moscow, GRMFML, 1976 10 Ozdeger A. A Chebyshev nets in a subspace of a Weyl space and the generalized Dupin theorem. Webs and Quasigroups, 1995. 97-105 11 Kofoglu N, Ozdeger A. Some special nets of curves in a Weyl hypersurface. Webs and Quasigroups, 2000. 104-113 12 Zlatanov G, Tsareva B. On the Geometry of the nets in the n-dimensional space of Weyl. Journal of Geometry, 1990, 38: 182-197 13 Canfes E 0, Ozdeger A. Some applications of prolonged covariant differentiation in Weyl spaces. Journal of Geometry, 1997, 60: 7-16