Construction of a complete negatively curved singular Riemannian foliation

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J.KoreanMath.Soc.32 (1995), No. 3,pp. 609-814

CONSTIlUCTION OF A COMPLETE NEGATIVELY CuRVED SINGULAR RlEMANNIAN FOLIATION

HARVO KITAHAR.A ~ND aONG KVUNG PAX*

Let (M, g) be a complete Riemannia:n manifold and G be a closed (eonneeted) subgroup of the group of i80metries

of

M. Then the union

o '

M

of

all principal orbits is an open dense subset of M and the quotient

o 0 0 ,

map M

- +

B

:=

M IG becomes a Riemannian submersion fOl" the

o ⁰

restriction

of

9

to

M which gives the quotient metric on B. Namely, B is a singula:r (complete) Riemannian space such that aB consists

of

non-principal orbits.

We

shall

discuss a complete singular Riemannian foliation F

on.

a complete Riemann;an manifold (M,g) of dimension m ([4]) such that B

:=

M IF

^is

a Riemannian manifold of dimension

q

with boundary,

o

namely, B is the regula:r stratum and 8B consists of singular leaves.

'Thrminology "complete" means that M is complete as a metric

SPace.

In this situation,

we

can CODStntct a complete negatively curved singular Riemannianfoliation with leaves sm-'(l) from (M,g,F) with the additional conditions :

THEoREM.

Let J ^be

^a

Coo -function

on

B. Suppose tbat B and f

satisfy tbe folloW'ing conditions :

(B.l) the canonically endowed continuous metric of-tbe Riemannian simple double 2B is smooth,

(B.2) tbe sectional curvature KB of B

is

negative, namely, tbe transversal sectional curvature

of

F is negative, or B

:=

[0,00),

(F.l) f is a function

of

tbe geodesic distance r

from

lJB, namely, f

is a basic function of the transversal distance r,

ReeeiwdAugust 9,1994.

1991 AMS Subject Classification: 53C20.

Key words: singular Riemannian foliation, warped Product, completion.

·The present Studies were supported (in part) by the Basic Science Research Institute Program, Ministry of Education, 1994, Project No. BSRI-94-1404.

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(F.2) I is an odd function of

r

on

a

neighborhood of

r =

0 and satisfies .that /'(0).= 1,' p'(r) >.p for:r > 0, and !"'(r) > o..

Then

there~ts.acomplet~

strictly negatively curvedRiemannian

m~oid'

With sm'-q(l)

~.

genenc 'leaves. .

We shall be in COO -categoJ;y,an,d manifolds are assumed to be con- nected, paracmnpact,HauSdorlf spaces. .' •

1. Preliminaries

LEMMA 1.1.

(cf.

[3],

[5],

p31)

If 1ft) is

a

real-valued Coo-even function

on

R, then I(

r)

is

a

Coo-function on R

^m-^q^,

where

T :=

({X1)2 + ... + (X

m -^q

?)1/2.

LEMMA l-.2~"([2})

'Let'

!';Rq x'R.m.,,-q+1--+·R·be·'a continuous

function. If f satisfies the following conditions :

(1.2.1) I is of class Coo an (Rq x Rm-q+I)\(Rq x {O}),

(1.2.2) fis invariant under {Iq} x O( m -

q

+ 1), where Iq is the unit group onRQ and O{m...:.. q + 1)18 the rotation group on

_R~;...q+l,

(1.2.3) I is.of class Coo on Rt' xl fOTany straight line 1 C Rm-q+I tbro,ugh the origin, then I is of class Coo onRq x

Rm-Q+I .

We suppose that B and I satisfy,the following coIiditio:ns:

'.

(1) B has the Riemannian simple double 2B,

(2) I(x) > 0

if

x E B\aB, and I is an oddfUIletiononaneighbor- hOQd of· aB.

ofth~

'arc-Iength r in. the. inner normal· direction to aB,

(3) IIgraet fll(x) = 1 ifxE BB..

Let (U, t/J) be a local patch of aB arouIid a singular leaf whose: di- mension is less than

.-!llat~of

.the generic leaf. Let· N be the f:"collar neighborhood of U in B. We define a: manifold N by

Imbedding of sm-q (1) into R

^m

-q+l, we define adiffeomorphism '1' of

.N into Rq x Rm-q+l by

'1':

({x,exprX),y) -+(<,b(x),ry),

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Construction of a complete negatively 611

where X

E

TxB is the unit inner normal vector to 8B and 0 <

r

<

^E.

We take the Riemannian metric g' on 'iJ!(N) so that 'iJ! may be- come an isometry. Note that

g'

can be extended to the continu- ous metric g' on 'iJ!(N) which is the closure of 'iJ!(N) by the natu- ral way. We have only to show that g' is of class Coo at the ori- gin. Let (xl, ... ,x

^q^,

x

^q

+

¹,'"

,x

^m

+ 1) be the Cartesian coordinates of

Rq x

R

^m-q+l.

And we adopt the ranges of indices:

1 S

i,j

S

q

and

q

+ 1 S

0:,(3

S

m

+

l.

It is clear from Lemma

1.2

that g'ii

:=

g'(8j8x i ,8j8xi ) is of class Coo.

Moreover, we have g'iOf := g'(8j8x i ,8j8xa) = xa(1jr)g'(8j8x i ,8j8r) is of class Coo. Finally we

see

that using polar coordinates

g'~fJ:= ^g'(8jox ^Of ^{,ajaxfJ )}

- f2(x,r) -

r² 4 a

a ^fJ)

= gafJ +

⁴ ^r

gsm-q(a/ax , lax

r

- f

²

(x,r)-r

²⁽ ^2-

Of fJ)

=

gafJ +

⁴ ^r

gafJ - x x

r

where 9 is the standard metric on

Rqx

R m-q+!.

It

follows from Lemma

1

²(x r) r² -

1.2

that

~4 ^-

is of class Coo. Therefore, g' afJ is of class Coo.

Taking sm-q(l) in the tangent space to the generic leaf, we set

M

^:=

^(B\8B)

^x^IIB\8B

sm-

^g

^(1). Then there exists the unique complete singular Riemannian foliation on M with sm-

^q

(1) as leaves which is the completion of M.

Summing up, we have

0 0 0

PROPOSITION

1.3. Let M be the regular stratum and B

:=

M I:F.

Suppose that B and f satisfy the following conditions:

(1) B has the Riemannian simple double 2B,

(2) f(x) >

0

if x

E

B\aB, and f

is

an odd function on

a

neighbor- hood of aB of the aTe-length r in the inner nonnal direction

to aB,

(3)

IIgrad

fll(x) =

1if

x

E

aB.

Then there exists the unique complete singular Riemannian foliation

on M with sm-q (1) as generic leaves.

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LEMMA

1.4. ([1]) Let M

:=

B xI F be a warped product with a warping function j where B and F are any Riemannian manifolds. Let

11"1

.and

11"2

be the natural projections respectively. Let 11 be a 2-plane tangent to M at x and {X + ^{V, Y} + W} an orthonormal basis for 11, where

X,Y

E T

_1r1(x)B

and V; W E T

1r2(x)F.

The sectional curvature K(II) ofII in M is given

by

K(II) = Kly +Kiyvw+Ktw,

_, _, , , ,

where

K~

, y

:=

KB(X, Y)IIX

AYII~

K1,Y,v,w:= -j(1I"1(x)){IIWII}«VB)2f)(X,X)

- 2

< V, W ,;>p «VB)2 f)(.x:, Y) + ^HVII}«VB)2

f)(~

Y)}, Kt.-,w

:=

j2(1I"(x)){Kp(V, W) -lIgrad

jll~}IIV^A

WII},

and

Vo

and

KO

denote the covariant derivative and the sectional curvature of (.) respectively and (VB)2 j denotes the Hessian of j.

2. Proof of Theorem

By the conditions imposed on B, there is a diffeomorphism

'1i ;

aB x [0,(0)

--+

B such that, for any x E 8B,T

_x

(r}:= '1i(x,r) is the geodesic parametrized by the .arc-Iength r, starting at x and normal to aB. Since Lemma 2.1, (B.2) and (F.2) imply that K I, K2 and 1(3

are non-positive on M and at least one of K

^I,

K

²

and K

³

is strictly negative, it is enough to show that at least one .of K

^I,

K

²

and K

³

is strictly negative

if

r

~

O. Let

Xo

be a boundary point of B and X

r ,1';.,

V

r ,

W

r .

be :any vector fields along Txo(r), where X

r ,

Y

r

are transversal and V

r ,

W

r

are tangent to leaves

if

r #- o.

Case 1. The case that Xo and 1'0 are linearly independent. We have Kl

_o,Yo

< O.

Case 2. The case that Vo ^and Wo are linearly independent. (F.1 ) and (F.2) imply that

j2(r) = r

²

+ 2ar

⁴

+"', a> 0

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and

Then we have

so that

Construction of a complete negatively

IIgrad f(r)1I1

_~<

grad f(r),o/fJr >1

=

(O~~)r

=

1 + 6ar

²

+ ....

1 - IIgrad f(r)111 < 1 - (1 + 6ar

²

+ )

j2(r) - r

²

+ 2ar

⁴

+ .

_ -6a + o(r) - 1 + o(r) ,

lim Kt. w :s -6a < O.

r-.O r, r

613

Case 3. The case except Case 1 and Case 2. We can choose X n Y

_{r ,}

V

_n

W

_r

such that Y

_r = Cl

X r and l¥r

= C2

V

_n

where

Cl

and

C2

are constants with

^Cl

::j:.

^C2.

Let II

r

be the 2-plane spanned by the orthonormal basis {X

^r

+ Vr, Y

r

+ W

r }.

Then we have

To get limr_o K(II

_{r )}

< 0, it is enough to show that

under the assumption IIX

r

llB

⁼ 1.

and (F.2) imply the claim. Therefore we have the Theorem.

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References

1.R. L. Bishop and B. O'Neill, Manifolds of negative curvature, Traos. Amer.

Math. Soc. 145 (1969), 1-49.

2. H. Kitahara, H. Kawakami and J. S. Pak, On a construction of complete simply connected Riemannian manifolds with negative curvature,Nagoya Math.J.113 (1989), 7-13.

3.J. L. Ka.zdan and F. W. Warner, Curvature functions for open 2-manifolds, Ann. of Math. 99 (1974), 203-219.

4. P. Molino, Riemannian foliations, Birkhaiise Boston-Basel, 1988.

5. T. Ochiai and J. Noguchi, Geometric function theory (in Japanese), Iwanami, 1986.

Construction of a complete negatively curved singular Riemannian foliation

CONSTIlUCTION OF A COMPLETE NEGATIVELY CuRVED SINGULAR RlEMANNIAN FOLIATION

Let (M, g) be a complete Riemannia:n manifold and G be a closed (eonneeted) subgroup of the group of i80metries

M. Then the union

M

all principal orbits is an open dense subset of M and the quotient

map M

B

M IG becomes a Riemannian submersion fOl" the

restriction

9

M which gives the quotient metric on B. Namely, B is a singula:r (complete) Riemannian space such that aB consists

non-principal orbits.

We

discuss a complete singular Riemannian foliation F

a complete Riemann;an manifold (M,g) of dimension m ([4]) such that B

M IF

a Riemannian manifold of dimension

with boundary,

namely, B is the regula:r stratum and 8B consists of singular leaves.

'Thrminology "complete" means that M is complete as a metric

In this situation,

can CODStntct a complete negatively curved singular Riemannianfoliation with leaves sm-'(l) from (M,g,F) with the additional conditions :

Let J be

Coo -function

B. Suppose tbat B and f

satisfy tbe folloW'ing conditions :

(B.l) the canonically endowed continuous metric of-tbe Riemannian simple double 2B is smooth,

(B.2) tbe sectional curvature KB of B

negative, namely, tbe transversal sectional curvature

F is negative, or B

[0,00),

(F.l) f is a function

tbe geodesic distance r

lJB, namely, f

is a basic function of the transversal distance r,

(F.2) I is an odd function of

on

neighborhood of

0 and satisfies .that /'(0).= 1,' p'(r) >.p for:r > 0, and !"'(r) > o..

Then

strictly negatively curvedRiemannian

With sm'-q(l)

genenc 'leaves. .

We shall be in COO -categoJ;y,an,d manifolds are assumed to be con- nected, paracmnpact,HauSdorlf spaces. .' •

1. Preliminaries

(cf.

[5],

If 1ft) is

real-valued Coo-even function

R, then I(

is

Coo-function on R

where

({X1)2 + ... + (X

?)1/2.

'Let'

function. If f satisfies the following conditions :

(1.2.1) I is of class Coo an (Rq x Rm-q+I)\(Rq x {O}),

(1.2.2) fis invariant under {Iq} x O( m -

+ 1), where Iq is the unit group onRQ and O{m...:.. q + 1)18 the rotation group on

(1.2.3) I is.of class Coo on Rt' xl fOTany straight line 1 C Rm-q+I tbro,ugh the origin, then I is of class Coo onRq x

We suppose that B and I satisfy,the following coIiditio:ns:

(1) B has the Riemannian simple double 2B,

(2) I(x) > 0

x E B\aB, and I is an oddfUIletiononaneighbor- hOQd of· aB.

'arc-Iength r in. the. inner normal· direction to aB,

(3) IIgraet fll(x) = 1 ifxE BB..

Let (U, t/J) be a local patch of aB arouIid a singular leaf whose: di- mension is less than

.the generic leaf. Let· N be the f:"collar neighborhood of U in B. We define a: manifold N by

Imbedding of sm-q (1) into R

-q+l, we define adiffeomorphism '1' of

.N into Rq x Rm-q+l by

'1':

where X

TxB is the unit inner normal vector to 8B and 0 <

<

We take the Riemannian metric g' on 'iJ!(N) so that 'iJ! may be- come an isometry. Note that

can be extended to the continu- ous metric g' on 'iJ!(N) which is the closure of 'iJ!(N) by the natu- ral way. We have only to show that g' is of class Coo at the ori- gin. Let (xl, ... ,x

x

Let J ^be

g'~fJ:= ^g'(8jox ^Of ^{,ajaxfJ )}

a ^fJ)

^(B\8B)

^(1). Then there exists the unique complete singular Riemannian foliation on M with sm-

be the natural projections respectively. Let 11 be a 2-plane tangent to M at x and {X + ^{V, Y} + W} an orthonormal basis for 11, where

< V, W ,;>p «VB)2 f)(.x:, Y) + ^HVII}«VB)2