ON THE GEOMETRY OF SUBMERSIONS

INTERNATIONAL JOURNAL OF GEOMETRY
Vol. 3 (2014), No. 2, 51 - 56
ON THE GEOMETRY OF SUBMERSIONS
A. Ya. NARMANOV and A. S. SHARIPOV
Abstract. In this paper we study the geometry of the level surfaces of
functions of certain class. It is proved that the level surfaces of these functions generate a foliation whose leaves are manifolds of constant Gaussian
curvature.
1. Introduction
Let M be a smooth Riemannian manifold of dimension n with Riemannian
metric g, F -foliation of dimension k on M , where 0 < k < n [6]. We denote
by Lp - a leaf of the foliation F passing through the point p 2 M , by Tq F the tangent space of leaf Lp at the point q 2 Lp , and by H(q) - orthogonal
complement of Tq F: As a result, we get two sub-bundles T F = fTq F g;
T H = fH(q)g of the tangent bundle T M , and we have the orthogonal
decomposition T M = T F H. Thus every vector …eld X expanded in the
form X = X v + X h , where X v 2 T F , X h 2 T H. If X h = 0 (respectively
X v = 0), the …eld is called the vertical ( horizontal) vector …eld.
The Riemannian metric g on a manifold M induces Riemannian metric
ge on the leaf Lp . Canonical injection i : Lp ! M is an isometric immersion
with respect to these metrics. A connection r (Levi-Civita connection)
e on Lp which
de…ned by the Riemannian metric g induces a connection r
e [1].
coincides with the connection de…ned by the Riemannian metric r
Let Z be a horizontal vector …eld. For each vertical vector …eld X we
de…ne the vertical vector …eld
S(X; Z) = (rX Z)v :
and we get the tensor …eld of type (1,1)
X; Z ! SZ X = S(X; Z):
This tensor …eld gives the bilinear form lZ (X; Y ) = hSZ X; Y i; where hX; Y i
- the inner product de…ned by the Riemannian metric g.
————————————–
Keywords and phrases: riemannian manifold, connection, foliation,
leaf, hessian, submersion, Gauss curvature.
(2010)Mathematics Subject Classi…cation: 53C12, 57R30
Received: 03.12.2013. In revised form: 20.06.2014. Accepted: 17.07.2014.
52
A. Ya. Narmanov and A.S.Sharipov
Tensor …eld SZ is called the second fundamental tensor, and the form
lZ (X; Y ) is called the second main form with respect to the horizontal …eld
Z.
Mapping SZ : Tq F ! Tq F de…ned by the formula Xq ! S(X; Z)q is selfendomorphism with respect to the scalar product de…ned by the Riemannian
metric ge. If the vector …eld Z is a vector of unit length, the eigenvalues of
this endomorphism called the principal curvatures of Lp at the point q , and
the corresponding eigenvectors are called the principal directions.
The mean curvature HZ and the Gauss-Kronecker curvature KZ of the
leaf Lp at the point q are determined by the principal curvatures [1]:
HZ = k1 trSZ ,KZ = detSZ , where k = n m:
Let f : M ! B be a di¤erentiable map of maximal rank, where M; B a
smooth manifolds of dimensions respectively. Such maps are called submersions. By the theorem on the rank of a di¤erentiable function for each point
p 2 B inverse image f 1 (p) is a submanifold of dimension k = n m. Thus
submersion f : M ! B generates a foliation on a manifold of dimension
k = n m whose leaves are submanifolds Lp = f 1 (p); p 2 B.
The geometry and topology of the foliations generated by submersions
were subject of numerous studies [2],[4],[5],[6].
If the di¤erential df of mapping f : M ! B preserves the length of
horizontal vectors, the submersion f : M ! B is called the Riemannian
submersions. Geometry of Riemannian submersions investigated in numerous studies, in particular in [5] derived the fundamental equations of the
Riemannian submersion.
2. MAIN RESULT
We consider the case where the manifold B is one-dimensional manifold,
to be more precise we consider smooth function f : M ! R. If Critff g the set of critical points of the function f , then on the manifold M n Critff g
arises foliation F of dimension n 1 (or codimension one foliation), leaves
of which are level surfaces of function f .
In [6] it is studied the geometry of the level surfaces of the functions for
which X(jgradf j2 ) = 0 for each vertical vector …eld X. In particular, it is
proved that the level surfaces form the so-called Riemannian foliation.
In this paper, we show that the level surfaces of the function of this class
are the surfaces of constant Gaussian curvature.
Theorem 2.1. Let f : M ! R be a smooth function. Suppose that Critf =
? and X jgradf j2 = 0 for each vertical vector …eld X. Then every leaf of
foliation F (level surface of f ) is a manifold of constant Gaussian curvature.
Proof. As is known the Hessian is given by
hf (X; Y ) = lZ (X; Y ) = hrX Z; Y i
where Z = gradf , r- the Levi-Civita connection de…ned by Riemannian
metric g.
ON THE GEOMETRY OF SUBMERSIONS
53
The map X ! hf (X) = rX Z (Hesse tensor) is a linear operator and is
given by a symmetric matrix A:
hf (X) = rX Z = AX:
We denote by ( ) the characteristic polynomial of the matrix A with a
free term ( 1)n det A and de…ne a new polynomial ( ) by the equation
( ) = det A
( 1)n ( ):
Since (A) = 0 we have that A (A) = det A E , where E - is the identity
matrix. The elements of the matrix (A) are cofactors of the matrix A. This
matrix is denoted by Hfc .
It is well known that the Gaussian curvature of the surface is calculated
by the formula ([1], p.110)
K = det S =
1
c
n+1 Hf (gradf ) ; gradf :
jgradf j
To prove the theorem it su¢ ces to show that X (K) = 0 for each vertical
vector …eld X at any point q of a leaf Lp .
By hypothesis of the theorem we have
X jgradf j2 = 0
and so
X
therefore we need to show that
1
jgradf jn+1
=0
rX Hfc Z; Z + Hfc Z; rX Z = 0:
We know that if X(jgradf j2 ) = 0 for each vertical vector …eld X, each
gradient line of f is a geodesic line of Riemannian manifold [4]. By de…nition,
Z
the gradient line is a geodesic if and only if rN N = 0; where N = jZj
.
We calculate the covariant di¤erential
1
1 1
1
rN N =
rZ N =
(
rZ Z + Z(
)Z) = 0
jZj
jZj jZj
jZj
1
). This means that the gradient
and get rZ Z = Z, where = jZjZ( jZj
vector Z is the eigenvector of matrix A:
Let X10 ; X20 ;
; Xn0 1 ; Z 0 - be mutually orthogonal eigenvectors of A at
the point q 2 Lp such that X10 ; X20 ;
; Xn0 1 the unit vectors, Z 0 - the value
of the gradient …eld at a point q. Locally, they can be extended to the vector
…elds X1 ; X2 ;
; Xn 1 ; Z to a neighborhood of (say U ) point q so that they
formed at each point of an orthogonal basis consisting of eigenvectors. We
construct the Riemannian normal system of coordinates (x1 ; x2 ;
; xn ) in
a neighborhood U via vectors X10 ; X20 ;
; Xn0 1 ; Z 0 ([1], p.112).
The components gij of the metric g and the connection components kij
in the normal coordinate system satis…es the conditions of ([1], p. 132):
gij (q) =
ij ;
k
ij (q)
= 0:
54
A. Ya. Narmanov and A.S.Sharipov
We show that X( ) = 0 for each vertical …eld X. From the equality
X( )=
X (jZj) Z
1
jZj
jZj X Z
1
jZj
and from the condition X(jZj) = 0 follows equality
X Z
1
jZj
= X (Z ( )) = [X; Z] ( )
Z (X ( ; ))
1
where = jZj
, [X; Z]-Lie bracket of vector …elds X; Z.
From the condition of the theorem follows X (Z ( )) = 0. In [6] it is
shown that X jgradf j2 = 0 for each of the vertical vector …eld X if and
only if [X; Z] a vertical …eld. Therefore [X; Z] ( ) = 0. Thus, is a constant
function on the leaf L.
Now we denote by 1 ; 2 ; : : : ; n 1 the eigenvalues of the matrix A corresponding to the eigenvectors X1 ; X2 ; : : : ; Xn 1 . Then in the basis X1 ; X2 ; : : : ,
Xn 1 ; Z matrix A has the form:
0
1
0 ::: 0
1
B 0
0 C
2 :::
C
A=B
@ ::: ::: ::: ::: A
0 0 ::: n
By hypothesis of the theorem, the vector …eld rX Z is vertical …eld. It
follows Codazzi equations have the form ([3], p.29)
(rX A)Y = (rY A)X
From this equation we get
rXi AXj = rXj AXi ; rXi AZ = rZ AXi
(1)
at any point of U for each vector …eld Xi . From …rst equation of (1) we take
following equality
Xi ( j )Xj +
j rXi Xj
= Xj ( i )Xj +
i rXj Xi :
(2)
Since rXi Xj = kij Xk = 0 at the point q by properties of normal coordinate system, from (2) follows equality
Xi ( j )Xj = Xj ( i )Xj :
By the linear independence X1 ; X2 ;
; Xn
1,
(3)
we have that
Xi ( j ) = 0 for i 6= j for all i.
From second equation of (1) we take following
Xi ( )Z + rXi Z = Z( i )Xi +
i rZ X i :
(4)
Since
rXi Z = rZ Xi = 0
at the point q from the linear independence of vectors Xi ; Z we have that
Xi ( ) = 0; Z( i ) = 0 for all i.
ON THE GEOMETRY OF SUBMERSIONS
55
On the other hand
rZ AXi = Z( i )Xi +
From (5) we get that
2
i Xi
i rZ Xi ; rXi AZ
= rZ AXi ; rXi Z =
+ Z( i )Xi = Xi ( )Z +
i Xi :
i Xi :
(5)
(6)
2
i
Since Z( i ) = 0,Xi ( ) = 0 at the point q from the (6) follows that
=
and X( i ) =
i : In particular, this implies that if i 6= 0 then i =
X( ) = 0, Z( ) = Z( i ) = 0 for all i. Thus, in the neighborhood U of the
point q non-zero eigenvalues of the matrix A are constant and equal .
Given this fact we compute X (K). We denote by m the number of zero
eigenvalues of A. If m = 0, all the eigenvalues are equal to the number .
In this case, by the de…nition of the matrix Hfc we get that Hfc Z = n 1 Z.
Consider the case when m > 0. If m > 1, then Hfc = 0. If m = 1 than
i = 0 for some i and AXi = rXi Z = 0: This means that the vector …eld Z
is parallel along the integral curve of a vector …eld Xi (along i - coordinate
line). If i = n we have = i = 0 for all i and Hfc = 0:
Without loss of generality we assume that i < n: In this case vector Hfc Z
@
: In this case we get
have only one nonzero component bi and Hfc Z = bi @x
i
rX Hfc Z = X(bi )
@
@
+ bi r X
:
@xi
@xi
@
@
vertical and rX @x
= 0: Thus in the case m = 1
As we know that Xi = @x
i
i
E
D
we have rX Hfc (gradf ); gradf = 0
Let us consider the case m = 0: In this case we have equalities Hfc Z =
n 1
Z and
rX Hfc Z = X n 1 Z + n 1 rX Z:
As mentioned above …eld rX gradf is a vertical vector …eld for each vertical
vector …eld X (the
E …eld AX is vertical). From this equalities follows
D
c
rX Hf (gradf ); gradf = 0 at the point q: The theorem is proved.
Examples:
1. M = R3 n f(x; y; z) : x = 0; y = 0g,f (x; y; z) = x2 + y 2 . Level surfaces
of this submersion are manifolds of zero Gaussian curvature.
2. M = R3 n f(0; 0; 0),f (x; y; z) = x2 + y 2 + z 2 . Level surfaces of this
submersion are manifolds of constant positive Gaussian curvature.
References
[1] Gromoll, D., Klingenberg, W. and Meyer, W., Riemannsche Geometrie im Grossen,
Lecture notes in mathematics (Springer-Verlag-Berlin), 1968, 55.
[2] Hermann, R., A su¢ cient condition that a mapping of Riemannian manifolds to be a
…ber bundle, Proc. Amer. Math. Soc., 11(1960), 236-242.
[3] Kobayashi, Sh. and Nomizu, K., Foundations of di¤ erential geometry - Vol.2, Interscience Publishers, New York-London-Sydney, 1969, 485.
[4] Narmanov, A. and Kaipnazarova, G., Metric functions on Riemannian manifolds,
Uzbek Mathematical Journal, 1(2010), 112-120 (Russian).
56
A. Ya. Narmanov and A.S.Sharipov
[5] O’Neil, B., The Fundamental equations of a submersions, Michigan Mathematical
Journal, 13(1966), 459-469.
[6] Tondeur, Ph., Foliations on Riemannian manifolds, Springer-Verlag, New York, 1988.
DEPARTMENT OF GEOMETRY
NATIONAL UNIVERSITY OF UZBEKISTAN, UZBEKISTAN
E-mail address: [email protected]
DEPARTMENT OF GEOMETRY
NATIONAL UNIVERSITY OF UZBEKISTAN, UZBEKISTAN
E-mail address: [email protected]

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