SOME RESULTS ON A RIEMANNIAN SUBMERSION H. Mete

İstanbul Üniv. Fen Fak.
Mat.Fiz.Astro. Derg. 1 (2004-2005), 115-126
S O M E R E S U L T S ON A RIEMANNIAN S U B M E R S I O N
H. Mete TAŞTAN
Abstract
In this paper, we develop some well-known results given by
O'Neill
[6], Gray [3] and Escobales [1] and obtain a few new results by using them.
1. Introduction
Let M
and B be smooth Riemannian manifolds. A Riemannian
submersion ;r : M —» i? is a mapping of M onto B satisfying the
following axioms;
51. it has maximal rank;
that is, each derivative map it„ of it is onto. Hence, for each q G B ,
7T~
is a submanifold of M of dimension dim M —dimB where the
[
x
submanifolds 7t~
are calledfibersof M .A vector field on M is called
vertical if it is tangent to afiberand horizontal if orthogonal to fiber.
52. 7T preserves lengths of horizontal vectors.
t
Given a Riemannian submersion K'.M-^>B
we denote by v the vector
subbundle of TM defined by the foliation of M by the fibers of 7t. %
denote the complementary distribution of v in TM determined by the
metric on M.
Recall that if p e M where M is any manifold, then T M denotes
p
the tangent space of M at p. Following O'Neill [6] we define the tensor T
of type (1,2) for arbitrary vector fields E and F by
T F=hV vF
E
vE
+
vV hF
vE
where vE, HE, etc. denote the vertical and horizontal projections of the
vector field E. O'Neill has described the following three properties of the
tensor T:
(1) T is a skew-symmetric linear operator on a tangent space of M and
reversing horizontal and vertical subspaces.
E
(2) T = T ,
E
yE
that is; T is vertical.
(3) For vertical vector fields V and W,
T W = T V.
y
T is symmetric, i.e.,
W
In fact, along a fiber, T is the second fundamental form of the fiber provided
we restrict ourselves to vertical vector fields.
Now, we simply dualize the definition of T by reversing v and % define
the integrability tensor A as follows. For arbitrary vector fields E mdF,
AF
E
= hV vF
tlE
+
W hF
tlE
(1') A is a skew-symmetric operator on TM reversing the horizontal and
vertical subspaces.
E
(2') A
E
= A , that is; A is horizontal.
hE
(3') For X,Y horizontal A is alternating, i.e., A Y =
X
-A X.
Y
2. The properties of vertical and horizontal distributions
Lemma 2.1 The vertical distribution V '. TM -> v(TM)is
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involutive.
Proof. Let V, W e v(TM),
we must show that [V, W] e v(TM) that is,
ñ[V,W] = 0.
h[V,W^ = %V' W — fiV' V
V
W
where V is the Riemannian connection on
M. By the definition of T, HV W = T W and fN V
V
h[V,W] = T W-T V
v
V
W
= T V . Hence
W
= 0.
w
Definition. A basic vector field is a horizontal vector field X which is
/T—related to a vector field X on B, i.e., izJX =X ^
for all
t
p e
p
1tn
M.
Lemma 2.2 I f X and Fare basic vectorfieldson M, then
1. {xj) =
{x,x)o
7U
2. h [X, F] is basic and is 7T - related [ X , , 7» ]
3. # V 7 is basic and is # —related V * Y
X
x
t
where V* is the Riemannian connection on B. The proofs of these results
are found in O'Neill [6].
Lemma 2.3 Let Z. be a basic vector field on M corresponding Z. on B.
Suppose for a horizontal vector field X, (X,Z¡)
p
= (X,Z¡)
, for all such
{
Z¡ and for any p,p' e 7T~ (q) where ^ e 5 . Then n^X is a well-defined
vector field on B. In particular X is basic.
See R.H. Escóbales [1].
Lemma 2.4 Let X and 7 be horizontal vector fields, V and J^be vertical
vectorfields.Then each of the following holds:
(1) A Y =
x
±v[X,Y]
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(2) V W = T W + V'yW , where V denotes the Riemannian connection
along a fiber with respect to the induced metric.
(3) a)VyX = hVyX + TyX
b) I f X is basic, KV X = A V
(4) V V = A V + vV V
(5) V Y = HV Y + A Y
V
V
r
x
x
X
x
X
X
X
The proofs of these results are found in 0'Neill[6] and R.H. Escobales[l].
Corollary 2.1 If X and Y are basic vector fields and V is vertical, then
V(X,Y)=0.
Proof: V(X,Y)
= <V X,F> + (X,V Y)
F
= (tff X,Y)
V
y
+
(X,hV X)
v
Since X and Fare basic. From Lemma 4 3b) we have
{WyXJ)
+ (X,hV X)
= (A V,Y) +
(A X,X)
v
X
Y
Now, if we use ( l ' ) we have
= -(A Y,V)-(A X,V),
x
-A Y
X
by
Y
- AX
Y
the
property
(3')
of
A,
= 0 and it follows.
Corollary 2.2 Horizontal distribution is involutive if and only if A = 0.
Proof. => Suppose horizontal distribution is involutive, that is; for any
horizontal X and F , [ ^ , F ] is horizontal. We must show that A E = 0,
for any vector field E.
X
From Lemma 4 -1) A Y = —v[X,Y]=0.
Thus, A %E = 0.
X
X
On the other hand, for any horizontal Z ;
(A vE,Z)
= -(A Z,vE)
x
x
Finally A E
X
=A
x
= ^implies A vE
= 0, since it is horizontal.
x
(KE + vE) = A hE
x
+ A vE
x
= QimpliesA = 0.
<= Now, i f A is identically zero, for any horizontal X and F
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O = A Y = -v[X,
Y] implies [X, Y] is horizontal, that is; horizontal
X
distribution is involutive.
3. Covariant derivatives of T and A
Lemma 3.1 I f X and Fare horizontal and V and W are vertical, then
(a) ( V A )
K
(c) ( V T )
x
= -A,
w
=- T
y
yW
(b) ( V A )
A v y
(d) ( V T ) =
x
F
=
w
-A
-T
y
KxW
TyY
Proof. We will only prove (c) since the proofs of others are similar. Let E
be an arbitrary vector field on M. Then
( V ^ T ) E = V (T E) - T
y
TE
Y
x
Y
= 0,T (V E)
Y
= 0,
X
T
E
v A ) = HvA )
and it follows.
T
E
(E) - T (V E)
VxY
y
and
T
E
= *A
since T is vertical,
X
from
)
'
h e n c e
V
Lemma
T
E
( x )r
=
2.4-(5)
T
E
- AA )
Corollary 3.1 a) I f A is parallel, then A is identically zero, i.e., for
aWEeTM, (V A) = 0 implies A = 0 .
E
b) I f T is parallel, then T is identically zero, i.e., ( V T ) = 0 implies
£
T=0.
The proofs of these results can be found in R.H. Escóbales [1}
Lemma 3.2 I f X,Y,Z,H
are vertical, then
are horizontal vector fields and
(a) ( ( V ^ A ^ V,W) = < V,
A W)
- {T W,
X
A V)
V
X
(b) < ( V A ) V,W) = (A V,A W)
-
(A W,A V)
(<0 < ( V x A ) Z,H) - (A Z,
-
(A H,A Z)
x
r
y
X
X
Y
A H)
Y
119
X
X
Y
Y
U,V,W,F
(d) < ( V
I /
A) ^,F> = 0
F
Proof. We will only prove b) since the proofs of others are similar.
(V,A)
V =V
Y
Hence
((V A\
(A V)
X
(Y) - A
VYX
x
= V {A V,W)
Y
where, (A V,W)
- <A ^ (V),W) - <A
X
V
V
Y
Y
-<A
and
(A V),W)
Y
HV W
Y
X
(A V),W)
X
since A V
X
on
X
Y
the
other
is
hand
since A is skew-symmetric, thus the
= {A V,A W)
X
X
Y
= {A V,A W)
X
= A W,
X
(V Y),W)
Y
= 0 since A reverses the horizontal
Y
and vertical subspaces and (A V,V W)
horizontal
- <A
VXY
= 0 , ( A ^ (V),W)
Y
X
-(A (V),W)
Y
- (A V,V W)
Y
(V V)
Y
V,W) = (V (A V),W)
X
X
- A
Y
Y
result follows.
Lemma 3.3 I f X,Y,Z,H
are vertical, then
are horizontal vector fields and
(a) ((V T) Y,Z)
X
=(A Y,T Z)-(A Z,T Y)
Y
X
(b) ((V^X X,Y) = {T X,T Y)
V
(c) {(S/ T)
V
W,F)
Y
V
V
-
V
= {T W,T F)
(d) {{y T) Z,H)
x
U,V,W,F
X
V
{T X,TJ)
V
- ( T ^ F , T W)
V
V
=Q
y
Proof. We will only prove a) since the proofs of others are similar.
( V T ) Y = V (TyY) - T
X
V
x
VxV
(Y) - T
(V Y)
v
X
Hence
((V T)
x
Y,Z) = (V (T>7),Z> - < T . (Y),Z)
y
x
= V (T Y,Z)
X
V
VA
- (T Y,V Z)
V
X
- (T
VxV
120
K
(Y),Z)-
- <Ty
(T
V
{V Y),Z)
X
(A Y),Z)
X
where (T Y,Z)
= 0,(T ^ F,Z) = 0 since T reverses the vertical and
V
v
K
horizontal subspaces and
(T Y, V Z) = (T Y, A Z)
V
vertical and v ( V Z ) = A Z
X
X
X
V
X
since T Y
y
is
from Lemma 2.4-(5). On the other hand
X
—(T ( A F ) , Z ) = (T Z, A Y)
v
V
since T is skew-symmetric, thus it
X
follows.
Lemma 3.4 I f X and Y are horizontal, and V and W are vertical, then;
(a) < ( V A ) Y , V ) is alternate in X and Y.
E
x
(b) < ( V T ) W , X )
E
v
is symmetric in V and W.
Proof. Expand the covariant derivatives and use the properties of T and A.
Lemma 3.5 I f V is vertical and i2 denotes the cyclic sum of over the
horizontal vector fields X, Y, Z, then
a((V A) Y,V)=Q(A Y,TyZ)
z
x
x
Proof. See O'Neill [6].
Corollary 3.2 I f U, V and W are vertical and Q. denotes the cyclic sum
of over U, V and W, then
i2(v(V T) ff) = 0
c /
K
Proof. For any vertical vector field F, we compute that
<KT)
K
W +
(V^TX V + (V T)
y
w
U,F) and applying Lemma 3.3-(c)
the result follows.
Corollary 3.3 I f Q. denotes the cyclic sum of over the horizontal vector
fields X,Y,Z and H horizontal, then
121
Q < ( V A ) Z,H) =
x
2n(A Z,A H)
r
x
Y
Proof. The result easily follows from the from Lemma 3.2 (c).
4. Fundamental Equations
Let 9i denote the curvature tensor of M, and 9{* the curvature tensor of
B. Since there is no danger of ambiguity, we will denote the horizontal lift
of 9T by 91* as well. Following O'Neill [6] we set
where h. are horizontal vectors such that n„ (/?,) = h*.
Theorem 4.1 I f
horizontal, then
U,V,W,F
(a) (K W,F)
= (<k W,F)
(b) W W,X)
= ((VJ)
UV
UV
uv
V
are vertical vector fields and
- {T W,T F)
V
+
Y
W,X) -((V T)
V
V
where $H is the curvature tensor of the fiber 7T
X is
(TyWJvF)
W,X)
at p.
Proof. (These equations relate the geometry of M to those of the fibers
n" (#) ; they are clearly the Gauss and Codazzi equations of the fibers.)
The proof is the same as that of a single submanifold.
X
Theorem 4.2 If X, Y, Z, H are horizontal vector fields and V is vertical,
then
122
(a)
WxrZ,!!) = (W Z,H)
~ 2(A Y,A H)
XY
X
+ (A Z,A H)
Z
Y
X
+
(A X,A H)
-
(A X,T Y)
Z
Y
(b)
WxyZn = <(V A)^ Y,V) + (A Y,T Z)
Z
X
- (A Z,7 X)
V
Y
y
z
Proof. See O'Neill [6].
Theorem 4.3 If X and Fare horizontal vector fields, and V and W are
vertical, then
(a) (n Y,W) = <(V T) W,Y) + < ( V A ) Y , W )
-(T X,T Y)
+
(A V,A W)
(b) (K X,Y)
= ({V A)
Y,W) - ({V A)
Y,V) +
(A V,A W)
-(A W,A V)-(T X,7 Y)
+ <T^X,T F>
xy
X
V
K
F
W
m
v
X
X
x
Y
Y
w
V
X
x
W
X
Y
K
Proof. See O'Neill [6].
In the case of sectional curvature, the proofs of these theorems become
trivial. For the tangent vectors a and b (assumed to be linearly
independent), we will denote by P £the tangent plane which is spanned by
them.
fl
Corollary 4.1 Let n: M —> B be a submersion and let K ,K and ft
denote the sectional curvatures of M, B and the fibers respectively. If x
and y are horizontal vectors at a point of M and v and w are vertical,
then;
t
(V A W,V A W)
123
y
(2)
K[if
(x,x){v,v)
,3 ,(P ) = , . ( P „ , ) - ^ |
)
v
;
W
l
, e
r
e
^ , .
W
.
Proof. (The first equation above is the formulation of Gauss equation for
the fibers.) All of them follow from the following well-known equation;
(VAW,VAW)
We have obtained in Lemma 3.1, Lemma 3.2 and Lemma 3.3 the covariant
derivatives of the fundamental tensors T and A; that is, we have expressed
the tensors
(V/T)
F
and ( V A )
£
f
in T and A.
Now, we consider ( ( V T ) G,L).
Since it can be written two different types of vector fields, horizontal and
vertical, instead of each vector fields E,F,G, and L , it follows that we can
state ( ( V T ) G,L) in exactly sixteen different types; i.e., we can say the
£
£
covariant derivatives VT of T in sixteen different type. The eight of them
can be expressed by using Lemma 3.1, and the three one of others can be
expressed by using Lemma 3.3 in T and A. But , the other five types may
not be possible to write in T and A by using the Lemma 3.1 and Lemma
3.3. For instance, we can not state ((V T) W,Y)
in T and A. Similar
x
y
claims are also valid for the fundamental tensor A. In other ways, we can
not state ((V A) Y,W)
in T and A. However, we find a relation between
y
<( V T)
x
y
x
W, Y) and <(V A)
Theorem 4.4 I f V and
horizontal, then
v
x
Y, W), and state it in Theorem 4.4 below.
W vertical vector fields and X and Fare
124
< ( V T ) W,Y) - <(V T)
X
K
Y
y
W,X)
= - < ( V A ) Y,W) - < ( V h \
K
X
w
X,V)
Proof. From the first Bianchi Identity we have that
(M Y
XV
+ 9i X
+ K V, W) = 0
VY
YX
Hence we can write
<9i Y,W)
xy
+ (W X,W)
+ (M V,W)
VY
= 0.. .(1)
YX
where, we use the Theorem 4.3-(a). Now we have that
W Y,W)
XV
= ((V T\
W,Y) + < ( V A ) Y,W) - (T X,T Y)
X
K
x
...(2),
and by the symmetries of curvature tensor
(M X,W)
VY
= -WyyX,W)
V
{A V,A W
, we obtain
= - < ( V T ) W,X)
r
+
W
- <(V A)
K
K
y
X,W)
+(T Y,T X)-(A V,A W)...(3)
On the other hand, again, by using the symmetries of curvature tensor 5R,
we have that
V
{SR V,W)
YX
= -(iH. V,W)
XY
W
Y
X
= -(1R X,Y)
and use the Theorem 4.3-(b)
VW
we have
{K V,W)
YX
= - < ( V A ) Y,W) + ((V A)
F
+(A W,A V)
X
Y
x
w
+ (T X,T Y)
V
W
Y,V) -
(A V,A W)
- (T XJ Y)
...(4)
x
Putting (2),(3) and (4) in (1) the result follows.
125
W
V
X
Y
X
Y
References
[1] R.H. Escobales, Jr., Riemannian submersions with totally geodesic
fibers, J. Differential Geometry 10 (1975) 253-276.
[2] K.L. Duggal and A. Bejancu, Lightlike submanifolds of SemiRiemannian manifolds and applications, Kluwer Academic Publishers,
1996.
[3]
A.
Gray, Pseudo-Riemannian almost product
manifolds and
submersions,}. Math. Mech. 16 (1967), 715-737.
[4] S. Kobayashi and K. Nomizu, Foundations of Differential Geomeùy,
Vol 1, Interscience, Newyork (1963).
[5] B. O'Neill, Semi-Riemannian Geometry with Applications to Relativity,
Academic Press, 1983.
[6] B. O'Neill, The Fundamental Equations of a Submersion, Michigan
Math. J. 13 (1966), 459-469.
H. Mete TAŞTAN
Department of Mathematics
Faculty of Science
Istanbul University
Vezneciler 34459, Istanbul
TURKEY
E-Mail: [email protected]
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