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Notes on infinitesimal variations of submanifolds
J. Math. SOC. Japan vol. 32, No. 1, lY80
Notes on infinitesimal variations of submanifolds By Kentaro YANO (Received April 1, 1978) § 0.
Introduction.
In a previous paper [ 5 ] , the present author studied variations of the metric tensor, the Christoffel symbols and the second fundamental tensors of submanifolds of a Riemannian manifold under infinitesimal variations of the submanifolds. In this paper, we assume that submanifolds under consideration are compact and orientable and we obtain, using integral formulas, some global results on infinitesimal isometric, affine and conformal variations of the submanifolds.
3 1. Preliminaries [l]. We consider an m-dimensional Riemannian manifold A I m covered by a system of coordinate neighborhoods { U ;x h } and denote by g,,, and V, the metric tensor, the Christoffel symbols formed with g,, and the operator of A i m respectively, where, of covariant differentiation with respect to here and in the sequel, the indices h , 1, j , k , ... run over the range {l',2', ... , m'}. We then consider an n-dimensional compact orientable Riemannian manifold A i " covered by a system of coordinate neighborhoods { V ;y"} and denote by g c b , v,, K d c b " and I ( c b the metric tensor, the Christoffel symbols formed with gcb, the operator of covariant differentiation with respect to c h , the curvature tensor and the Ricci tensor of Ad" respectively, where, here and in the sequel, the indices a, 6,c, run over the range 11, 2, ... , ? I } . We assume that hin is isometrically immersed in AIm by the immersion: hI"-Al" and represent the immersion by
r:,
r:,
Xh=X"(ya).
Since the immersion is isometric, we have
(1.1)
gcb=Bc'BbigJt9
where we have put BcJ=dcxJ (ac=a/ayc). We can assume that [ B b h ] gives the positive orientation of hi".
345
K. YANO
46
We choose m--n mutually orthogonal unit normals C V hto M", where, here and in the sequel, the indices x, y, z run over the range { n f l , ... , m } . The metric tensor of the normal bundle of M" is given by
(1.2)
gzy=CzjCyigji. Now, the equations of Gauss for hf" are written as
where vc Bb h=
acB bh
+r j l B
Bbt- r,",B a
is the van der Waerden-Bortolotti covariant derivative of Bbh and hcbZ are components of the second fundamental tensor with respect to the normal C Z h . On the other hand, the equations of Weingarten for Ail" are written as
(1.4) where
VcCyh=-hcayBah,
vcc,h=accyh+r;& B c~cyt -r;vc,h
r&
is the van der Waerden-Bortolotti covariant derivative of C,", being components of the linear connection induced in the normal bundle, that is,
rAJ=(a,cy+r:L B c ~ i)c= cy h
h
and Cxh=CyEgyagph,g y x being contravariant components of the metric tensor of the normal bundle of 121" and hcay=hcbzgbagzy, gba being contravariant components of the metric tensor of
M".
$ 2 . Infinitesimal variations C23 [ S ] [ 5 ] . We now consider an infinitesimal variation of hP given by
(2.1)
Xh=xh+E"y)&,
where Eh(y) is a vector field defined along M" and E is an infinitesimal. Under the infinitesimal variation (2.1) the vectors B b h tangent to h i'" are transformed into Bbh=abzh=Bbh+abEh& tangent to the deformed submanifold. Carrying B b h at ( f h ) back to ( x h ) parallelly, we obtain gbh=Bbh+r3i(
X+E&)E'Bbt&
that is,
346
,
Infinitesimal variations
47
neglecting terms of order higher than one with respect to
E,
where
In the sequel we always neglect the terms of order higher than one with respect to E . Thus putting o"Rbh=E b l l - Bbh, we have (2.4)
aI!?","=(V,f")&.
If we decompose (2.5)
thas ;'"=:" B , +E"C,h ,
equation (2.4) can be written as
where Vb=gbaV, and libax=hedrgEbgda. When 6gc6=0, we say that the infinitesimal variation is isometric and when dgcb=2Rpcbs,R being a certain scalar, we say that the infinitesimal variation is conforiiial. If the variation is conformal and R is a constant, we say that the infinitesimal variation is Iionzothetic. From (2.7) we have THEOREM A. [5] h i order f o r a n infinitesimal variation (2.1) o f a submanifold to be isometric, it is necessary and sujficieizt that
(2.9)
+
V c b VbtC-2 h C b X [ " =0
.
THEOREM B. [5] 112 order for a n infinitesimal variation (2.1) o f a submanif o l d to be conformal, it i s necessary and suficient that
(2.10)
VcEbtVb:c-2}2cbxE3=2Rgcb
9
A being a certain scalar. Using (2.7) and (2.8), we calculate the infinitesimal variation of the
347
K. YANO
48
Christoffel symbols 1
r?b=2(acgbe+ abgce-aegcb)gea and obtain (2.12)
6 l ' ~ b =1, C v c ( a g b e ) + v b ( s g ~ , ) - ~ e ( ~ ~ c b* ) l ~ ~ "
which, using (2.7), we car, write as
(2.13)
6fibb=CVcVbEa+ K d ~ b ' [ ~ - V c (hbaz[")-Vb( hc"xE")+V"( hcbzez)]s
-
If 6r,4,,=0, we say that the infinitesimal variation is affine and if =(6:pb+8$pc)& for some 1-form PO, we say that the variation is projective. T h u s we have THEOREM C. [ 5 ] In order for an infiiiitesiinal variation (2.1) o f a subinanifold to he af/ine, it is necessary and suficient that
(2.14)
+
VcVbE" KdCb"td-Vc( h b a z E z ) - v b ( hc"z[")+
vQ(/lb x E x ) = O L
*
Now we have the following integral formula C41
which is valid for an arbitrary vector field E" in a compact orientable Riemannian manifold M", dV being the volume element of the manifold. From (2.15), we can easily derive
1
c-211CbZE3)
+ ( v c ~ b + v b ~ c - ~ ~ c b ~ ~ " ) h c b ~ ~ z
-(Vc;c-hccxE")(Vb~b)]dI/=O,
which is valid for arbitrary and Ex. Now suppose that an infinitesimal variation (2.1) of the submanifold is isometric. Then since it is affine, we have (2.14), from which, we have, by transvection with gcb,
(2.17)
gcbvcVbEaf Kd"Ed-2Vc( h,",E")+ V a (h C c , ~ " ) = O .
We also obtain from (2.9),
348
Injnitesimal variations
(2.18)
49
(VCE~+V~EC-~~C~~E')~~~"E"=~
and (2.19)
(VcEC-
h ccz[J)(vb[b)
=o .
Conversely if (2.17), (2.18) and (2.19) are satisfied, we have, from (2.16), VcEbfVb~~-2~~cby[~=~ t which shows that the infinitesimal variation is an isometry. Thus we have
THEOREM D. [ 5 ] I n order f o r a n infinitesimal variation o f a coinpact orientable submanifold o f a Riemannian manifold to be isometric, it is necessary and sziflcient that we have (2.17), (2.18) and (2.19). $ 3 . Infinitesimal isometries. Suppose that the infinitesimal variation (2.1) is an isometry. have (2.17). T h u s substituting (2.17) into
from which, by integration over
M",
or
Since (2.1) is an isometry, we have
Thus substituting these into the above equation, we find
From (3.2), we have
349
Then w e
K. YAso
50
THEOREM 3.1. I f an iizfiiiitesimal isometric variation o f a compact orientable submanifold M" of a Riemannian inanifold M" satisjies (3.3)
~ ~ c b E c ~ b + 2 ( ~ ~ c b " , c b x9 ) ~ y ~ x ~ ~
6" satisfies
then
VcEa=O and cosequently KcbEcEb=O and hcb.dx=O, that is, IW is geodesic in the direction E x . Moreover i f izil" is irreducible, then ["=O, that is, the variation is normal and the submanifold is geodesic i n the direction o f the variation.
8 4. Infinitesimal affine variations. For an infinitesimal affine variation (2,1), we have (2.14), from which, by transvection with gcb, we obtain
~cbVc~~~afK~a~d-~VC(hcaS~")+Va(hcc,~~)=O
(4.1)
and, by contraction with respect to a and b, we have (4.2)
Vc(V,E" - h,",~")=O
and consequently we obtain (4.3)
VaEa- haa,Ex=constant. Thus (2.16) becomes ~
[1 ~
(
~
~
~
~
+
~
~
~
~
-
Vbfc-2 h c b y Ey=o
.
then we have, from the equation above, (4.5)
0cfb-t
The converse being evident, we have
3 50
~
h
c
b
y
~
~
)
(
~
~
~
~
~
~
~
Injnitesimal variations
51
THEOREM 4.1. I f an infinitesiinal aflne variation o f a compact orientable submanifold o f a Rieiizannian manifold satisfies (4.4)then the variation i s a n isornet ry.
Q 5. Infinitesimal conformal variations. If an infinitesimal variation (2.1) is conformal, we have (5.1)
vctb+v b t c -
2 h cbtt" =21gcb
for a certain function I , from which, transvecting with gcb, we find (5.2)
1 I =-(VaEQ 71
-
h .",~").
Thus we can write (5.1) a s
n-2 +--Vn(VeEen
heez[")=O ,
Now, we can transform (2.16) into
351
K. YANO
52
Thus if the infinitesimal variation is conformal, we have (5.3) and consequently (5.8) ' ( V c E b f v b ~ ~ - 2 h c b 2y ~ ~ - ~ ( /Iee,EV)gc*}/~",i"=o V e ~ ~ and also (5.6). Conversely if (5.6) and (5.8) are satisfied, we have from (5.7)
2
V e ~ b f V b ~ ~ - 2 ~ ~ c b y ~ ~ - - ( V e E ~ - h e ~ y E * ) g c tb = O 11
which shows that the infinitesimal variation is conformal. Thus we have THEOREM. 5.1. In order f o r an infinitesimal variation (2.1) to be conformal, it is necessary and szificient that (5.6) and (5.8) hold. Substituting (5.6) into (3.1), we find
1 -2 ~ ( $ E U ) = - ~ C ~ ~ ~ ~ ~ + ~ ~ Q V ~ ( ~ C " X E " ) - ~ U ~ ~ ( ~ C ~
_ _n -2 - EuVa(Ve4e-heexSx)+(VcEb)(VcSb) n
*
from which, integrating over M",
![
-Ir,*EcEb-(hcb",E")(vc~b+V,E,)f
hcCx4"(vaEu)
+~ ~ ( ~ ~ E ' - h ~ e , ~ " ) ( ~ Q E Q ) + ( ~ C ~ b ) ( ~ C ~ b ) ] d ~ = o Since the variation is conformal, we have
2 n
VrSb+vb~c=2hcbxEx+-(VeEe-
heedX)gcb
Substituting this into the above integral formula, we find
![ -KcbEESb-2hrbgE'
2 n
hcbxf"--hccy,EY(V,E"-Ilee~EZ)+
hc'&"(vaE")
+ "i (VeEe--hee~,E")(v~EQ)+(aP")(VcEb)]dV=O, "
352
Injnitesimal variations
from which (5.9)
Thus from (5.9), we have THEOREM 5.2. I f a n infinitesimal conformal variation of a coinpact orientable subiiianifold M" of a Rieiiiannian inaiiifold M" satisjes (5.10)
that is, M" is unibilical i n the direction i". Moreover if i\.I" is irreducible, then Ea=O, that is, the variation is normal a n d the szibrnanifold is unibilical in the direction o f t h e variation. Bibliography [ 1 ] Bang-yen Chen, Geometry of Submanifolds, Marcel Dekker, Inc., 1975. [ 2 ] J. A. Schouten, Ricci-Calculus, Springer-Verlag, 1954. [ 3 1 K. Yano, S u r la thkorie des dkformations infinitksimales, J. Fac. Sci. Univ. Tokyo, 6 (1919), 1-75. [4] K. Yano, Integral formulas in Riemannian geometry, Marcel Dekker, Inc., 1970. [ 5 1 K. Yano, Infinitesimal variations of submanifolds, Kodai Math. J., I (1978), 30-44.
Kentaro YANO Tokyo Institute of Technology Meguro-ku, Tokyo 152 Japan
353
Notes on infinitesimal variations of submanifolds By Kentaro YANO (Received April 1, 1978) § 0.
Introduction.
In a previous paper [ 5 ] , the present author studied variations of the metric tensor, the Christoffel symbols and the second fundamental tensors of submanifolds of a Riemannian manifold under infinitesimal variations of the submanifolds. In this paper, we assume that submanifolds under consideration are compact and orientable and we obtain, using integral formulas, some global results on infinitesimal isometric, affine and conformal variations of the submanifolds.
3 1. Preliminaries [l]. We consider an m-dimensional Riemannian manifold A I m covered by a system of coordinate neighborhoods { U ;x h } and denote by g,,, and V, the metric tensor, the Christoffel symbols formed with g,, and the operator of A i m respectively, where, of covariant differentiation with respect to here and in the sequel, the indices h , 1, j , k , ... run over the range {l',2', ... , m'}. We then consider an n-dimensional compact orientable Riemannian manifold A i " covered by a system of coordinate neighborhoods { V ;y"} and denote by g c b , v,, K d c b " and I ( c b the metric tensor, the Christoffel symbols formed with gcb, the operator of covariant differentiation with respect to c h , the curvature tensor and the Ricci tensor of Ad" respectively, where, here and in the sequel, the indices a, 6,c, run over the range 11, 2, ... , ? I } . We assume that hin is isometrically immersed in AIm by the immersion: hI"-Al" and represent the immersion by
r:,
r:,
Xh=X"(ya).
Since the immersion is isometric, we have
(1.1)
gcb=Bc'BbigJt9
where we have put BcJ=dcxJ (ac=a/ayc). We can assume that [ B b h ] gives the positive orientation of hi".
345
K. YANO
46
We choose m--n mutually orthogonal unit normals C V hto M", where, here and in the sequel, the indices x, y, z run over the range { n f l , ... , m } . The metric tensor of the normal bundle of M" is given by
(1.2)
gzy=CzjCyigji. Now, the equations of Gauss for hf" are written as
where vc Bb h=
acB bh
+r j l B
Bbt- r,",B a
is the van der Waerden-Bortolotti covariant derivative of Bbh and hcbZ are components of the second fundamental tensor with respect to the normal C Z h . On the other hand, the equations of Weingarten for Ail" are written as
(1.4) where
VcCyh=-hcayBah,
vcc,h=accyh+r;& B c~cyt -r;vc,h
r&
is the van der Waerden-Bortolotti covariant derivative of C,", being components of the linear connection induced in the normal bundle, that is,
rAJ=(a,cy+r:L B c ~ i)c= cy h
h
and Cxh=CyEgyagph,g y x being contravariant components of the metric tensor of the normal bundle of 121" and hcay=hcbzgbagzy, gba being contravariant components of the metric tensor of
M".
$ 2 . Infinitesimal variations C23 [ S ] [ 5 ] . We now consider an infinitesimal variation of hP given by
(2.1)
Xh=xh+E"y)&,
where Eh(y) is a vector field defined along M" and E is an infinitesimal. Under the infinitesimal variation (2.1) the vectors B b h tangent to h i'" are transformed into Bbh=abzh=Bbh+abEh& tangent to the deformed submanifold. Carrying B b h at ( f h ) back to ( x h ) parallelly, we obtain gbh=Bbh+r3i(
X+E&)E'Bbt&
that is,
346
,
Infinitesimal variations
47
neglecting terms of order higher than one with respect to
E,
where
In the sequel we always neglect the terms of order higher than one with respect to E . Thus putting o"Rbh=E b l l - Bbh, we have (2.4)
aI!?","=(V,f")&.
If we decompose (2.5)
thas ;'"=:" B , +E"C,h ,
equation (2.4) can be written as
where Vb=gbaV, and libax=hedrgEbgda. When 6gc6=0, we say that the infinitesimal variation is isometric and when dgcb=2Rpcbs,R being a certain scalar, we say that the infinitesimal variation is conforiiial. If the variation is conformal and R is a constant, we say that the infinitesimal variation is Iionzothetic. From (2.7) we have THEOREM A. [5] h i order f o r a n infinitesimal variation (2.1) o f a submanifold to be isometric, it is necessary and sujficieizt that
(2.9)
+
V c b VbtC-2 h C b X [ " =0
.
THEOREM B. [5] 112 order for a n infinitesimal variation (2.1) o f a submanif o l d to be conformal, it i s necessary and suficient that
(2.10)
VcEbtVb:c-2}2cbxE3=2Rgcb
9
A being a certain scalar. Using (2.7) and (2.8), we calculate the infinitesimal variation of the
347
K. YANO
48
Christoffel symbols 1
r?b=2(acgbe+ abgce-aegcb)gea and obtain (2.12)
6 l ' ~ b =1, C v c ( a g b e ) + v b ( s g ~ , ) - ~ e ( ~ ~ c b* ) l ~ ~ "
which, using (2.7), we car, write as
(2.13)
6fibb=CVcVbEa+ K d ~ b ' [ ~ - V c (hbaz[")-Vb( hc"xE")+V"( hcbzez)]s
-
If 6r,4,,=0, we say that the infinitesimal variation is affine and if =(6:pb+8$pc)& for some 1-form PO, we say that the variation is projective. T h u s we have THEOREM C. [ 5 ] In order for an infiiiitesiinal variation (2.1) o f a subinanifold to he af/ine, it is necessary and suficient that
(2.14)
+
VcVbE" KdCb"td-Vc( h b a z E z ) - v b ( hc"z[")+
vQ(/lb x E x ) = O L
*
Now we have the following integral formula C41
which is valid for an arbitrary vector field E" in a compact orientable Riemannian manifold M", dV being the volume element of the manifold. From (2.15), we can easily derive
1
c-211CbZE3)
+ ( v c ~ b + v b ~ c - ~ ~ c b ~ ~ " ) h c b ~ ~ z
-(Vc;c-hccxE")(Vb~b)]dI/=O,
which is valid for arbitrary and Ex. Now suppose that an infinitesimal variation (2.1) of the submanifold is isometric. Then since it is affine, we have (2.14), from which, we have, by transvection with gcb,
(2.17)
gcbvcVbEaf Kd"Ed-2Vc( h,",E")+ V a (h C c , ~ " ) = O .
We also obtain from (2.9),
348
Injnitesimal variations
(2.18)
49
(VCE~+V~EC-~~C~~E')~~~"E"=~
and (2.19)
(VcEC-
h ccz[J)(vb[b)
=o .
Conversely if (2.17), (2.18) and (2.19) are satisfied, we have, from (2.16), VcEbfVb~~-2~~cby[~=~ t which shows that the infinitesimal variation is an isometry. Thus we have
THEOREM D. [ 5 ] I n order f o r a n infinitesimal variation o f a coinpact orientable submanifold o f a Riemannian manifold to be isometric, it is necessary and sziflcient that we have (2.17), (2.18) and (2.19). $ 3 . Infinitesimal isometries. Suppose that the infinitesimal variation (2.1) is an isometry. have (2.17). T h u s substituting (2.17) into
from which, by integration over
M",
or
Since (2.1) is an isometry, we have
Thus substituting these into the above equation, we find
From (3.2), we have
349
Then w e
K. YAso
50
THEOREM 3.1. I f an iizfiiiitesimal isometric variation o f a compact orientable submanifold M" of a Riemannian inanifold M" satisjies (3.3)
~ ~ c b E c ~ b + 2 ( ~ ~ c b " , c b x9 ) ~ y ~ x ~ ~
6" satisfies
then
VcEa=O and cosequently KcbEcEb=O and hcb.dx=O, that is, IW is geodesic in the direction E x . Moreover i f izil" is irreducible, then ["=O, that is, the variation is normal and the submanifold is geodesic i n the direction o f the variation.
8 4. Infinitesimal affine variations. For an infinitesimal affine variation (2,1), we have (2.14), from which, by transvection with gcb, we obtain
~cbVc~~~afK~a~d-~VC(hcaS~")+Va(hcc,~~)=O
(4.1)
and, by contraction with respect to a and b, we have (4.2)
Vc(V,E" - h,",~")=O
and consequently we obtain (4.3)
VaEa- haa,Ex=constant. Thus (2.16) becomes ~
[1 ~
(
~
~
~
~
+
~
~
~
~
-
Vbfc-2 h c b y Ey=o
.
then we have, from the equation above, (4.5)
0cfb-t
The converse being evident, we have
3 50
~
h
c
b
y
~
~
)
(
~
~
~
~
~
~
~
Injnitesimal variations
51
THEOREM 4.1. I f an infinitesiinal aflne variation o f a compact orientable submanifold o f a Rieiizannian manifold satisfies (4.4)then the variation i s a n isornet ry.
Q 5. Infinitesimal conformal variations. If an infinitesimal variation (2.1) is conformal, we have (5.1)
vctb+v b t c -
2 h cbtt" =21gcb
for a certain function I , from which, transvecting with gcb, we find (5.2)
1 I =-(VaEQ 71
-
h .",~").
Thus we can write (5.1) a s
n-2 +--Vn(VeEen
heez[")=O ,
Now, we can transform (2.16) into
351
K. YANO
52
Thus if the infinitesimal variation is conformal, we have (5.3) and consequently (5.8) ' ( V c E b f v b ~ ~ - 2 h c b 2y ~ ~ - ~ ( /Iee,EV)gc*}/~",i"=o V e ~ ~ and also (5.6). Conversely if (5.6) and (5.8) are satisfied, we have from (5.7)
2
V e ~ b f V b ~ ~ - 2 ~ ~ c b y ~ ~ - - ( V e E ~ - h e ~ y E * ) g c tb = O 11
which shows that the infinitesimal variation is conformal. Thus we have THEOREM. 5.1. In order f o r an infinitesimal variation (2.1) to be conformal, it is necessary and szificient that (5.6) and (5.8) hold. Substituting (5.6) into (3.1), we find
1 -2 ~ ( $ E U ) = - ~ C ~ ~ ~ ~ ~ + ~ ~ Q V ~ ( ~ C " X E " ) - ~ U ~ ~ ( ~ C ~
_ _n -2 - EuVa(Ve4e-heexSx)+(VcEb)(VcSb) n
*
from which, integrating over M",
![
-Ir,*EcEb-(hcb",E")(vc~b+V,E,)f
hcCx4"(vaEu)
+~ ~ ( ~ ~ E ' - h ~ e , ~ " ) ( ~ Q E Q ) + ( ~ C ~ b ) ( ~ C ~ b ) ] d ~ = o Since the variation is conformal, we have
2 n
VrSb+vb~c=2hcbxEx+-(VeEe-
heedX)gcb
Substituting this into the above integral formula, we find
![ -KcbEESb-2hrbgE'
2 n
hcbxf"--hccy,EY(V,E"-Ilee~EZ)+
hc'&"(vaE")
+ "i (VeEe--hee~,E")(v~EQ)+(aP")(VcEb)]dV=O, "
352
Injnitesimal variations
from which (5.9)
Thus from (5.9), we have THEOREM 5.2. I f a n infinitesimal conformal variation of a coinpact orientable subiiianifold M" of a Rieiiiannian inaiiifold M" satisjes (5.10)
that is, M" is unibilical i n the direction i". Moreover if i\.I" is irreducible, then Ea=O, that is, the variation is normal a n d the szibrnanifold is unibilical in the direction o f t h e variation. Bibliography [ 1 ] Bang-yen Chen, Geometry of Submanifolds, Marcel Dekker, Inc., 1975. [ 2 ] J. A. Schouten, Ricci-Calculus, Springer-Verlag, 1954. [ 3 1 K. Yano, S u r la thkorie des dkformations infinitksimales, J. Fac. Sci. Univ. Tokyo, 6 (1919), 1-75. [4] K. Yano, Integral formulas in Riemannian geometry, Marcel Dekker, Inc., 1970. [ 5 1 K. Yano, Infinitesimal variations of submanifolds, Kodai Math. J., I (1978), 30-44.
Kentaro YANO Tokyo Institute of Technology Meguro-ku, Tokyo 152 Japan
353