.

.

.

...

.

Curvatures of Tangent Hyperquadric Bundles Takamichi Satoh Tohoku University

August 31, 2010 Joint work with Masami Sekizawa Takamichi Satoh (Tohoku Univ.)

Tangent Hyperquadric Bundles

August 31, 2010

1 / 24

Introducton

Introduction

Takamichi Satoh (Tohoku Univ.)

Tangent Hyperquadric Bundles

August 31, 2010

2 / 24

Introducton

THB

(M, g): pseudo-Riemannian manifold g¯: Sasaki metric on the tangent bundle T M over M Let r > 0,  = ±1. The tangent hyperquadric bundle (THB) of radius r over (M, g) is { } Tr M := (x, u) ∈ T M | gx (u, u) = r2 . We induce a pseudo-Riemannian metric g˜r on Tr M from the Sasaki metric g¯. e g r )(x,u) : scalar curvature of THB at (x, u) ∈ T  M. Sc(˜ r

Takamichi Satoh (Tohoku Univ.)

Tangent Hyperquadric Bundles

August 31, 2010

3 / 24

Introducton

THB

(M, g): pseudo-Riemannian manifold g¯: Sasaki metric on the tangent bundle T M over M Let r > 0,  = ±1. The tangent hyperquadric bundle (THB) of radius r over (M, g) is { } Tr M := (x, u) ∈ T M | gx (u, u) = r2 . We induce a pseudo-Riemannian metric g˜r on Tr M from the Sasaki metric g¯. e g r )(x,u) : scalar curvature of THB at (x, u) ∈ T  M. Sc(˜ r

Takamichi Satoh (Tohoku Univ.)

Tangent Hyperquadric Bundles

August 31, 2010

3 / 24

Introducton

THB

(M, g): pseudo-Riemannian manifold g¯: Sasaki metric on the tangent bundle T M over M Let r > 0,  = ±1. The tangent hyperquadric bundle (THB) of radius r over (M, g) is { } Tr M := (x, u) ∈ T M | gx (u, u) = r2 . We induce a pseudo-Riemannian metric g˜r on Tr M from the Sasaki metric g¯. e g r )(x,u) : scalar curvature of THB at (x, u) ∈ T  M. Sc(˜ r

Takamichi Satoh (Tohoku Univ.)

Tangent Hyperquadric Bundles

August 31, 2010

3 / 24

Introducton

THB

(M, g): pseudo-Riemannian manifold g¯: Sasaki metric on the tangent bundle T M over M Let r > 0,  = ±1. The tangent hyperquadric bundle (THB) of radius r over (M, g) is { } Tr M := (x, u) ∈ T M | gx (u, u) = r2 . We induce a pseudo-Riemannian metric g˜r on Tr M from the Sasaki metric g¯. e g r )(x,u) : scalar curvature of THB at (x, u) ∈ T  M. Sc(˜ r

Takamichi Satoh (Tohoku Univ.)

Tangent Hyperquadric Bundles

August 31, 2010

3 / 24

Introducton

theorem 1

e g r )(x,u) > 0 if  = 1, Sc(˜ e g r )(x,u) < 0 if  = −1. ... Sc(˜ . e g r )(x,u) : scalar curvature of THB at (x, u) ∈ Tr M. Sc(˜

Takamichi Satoh (Tohoku Univ.)

Tangent Hyperquadric Bundles

August 31, 2010

.

. . 1 .Theorem . (M, g), dim M ≥ 3 be a pseudo-Riemannian manifold with bounded sectional curvature. =⇒ For each sufficiently small r > 0,

4 / 24

Introducton

Theorem 2

e g r )(x,u) < 0 if  = 1, Sc(˜ e g r )(x,u) > 0 if  = −1. ... Sc(˜ . e g r )(x,u) : scalar curvature of THB at (x, u) ∈ T  M. Sc(˜

.

. . 2 .Theorem . (M, g), dim M ≥ 2 be a pseudo-Riemannian manifold of constant sectional curvature c 6= 0. =⇒ For each sufficiently large r > 0,

r

Takamichi Satoh (Tohoku Univ.)

Tangent Hyperquadric Bundles

August 31, 2010

5 / 24

Introducton

Corollary

.

Combining above results, we obtain the following. . . Corollary .. (M, g), dim M ≥ 3 be a pseudo-Riemannian manifold of constant sectional curvature c 6= 0. e g r ) ≡ c˜. ∀˜ c ∈ R, ∃r > 0 s.t. Sc(˜ ..=⇒ . . e g r )(x,u) : scalar curvature of THB at (x, u) ∈ Tr M. Sc(˜

Takamichi Satoh (Tohoku Univ.)

Tangent Hyperquadric Bundles

August 31, 2010

6 / 24

Definition of THB

Definition of THB

Takamichi Satoh (Tohoku Univ.)

Tangent Hyperquadric Bundles

August 31, 2010

7 / 24

Definition of THB

THB

where ...

Tr−1 M = ∅ Tr+1 M = ∅

if g is positive definite, if g is negative definite.

.

.

(M, g) : pseudo-Riemannian manifold . . .Definition . Let r > 0. The tangent hyperquadric bundle (THB) of radius r over (M, g) is defined by } { Tr M := (x, u) ∈ T M | gx (u, u) = r2

g is positive definite =⇒ Tr M = Tr M. Takamichi Satoh (Tohoku Univ.)

Tangent Hyperquadric Bundles

August 31, 2010

8 / 24

Definition of THB

THB

where ...

Tr−1 M = ∅ Tr+1 M = ∅

if g is positive definite, if g is negative definite.

.

.

(M, g) : pseudo-Riemannian manifold . . .Definition . Let r > 0. The tangent hyperquadric bundle (THB) of radius r over (M, g) is defined by } { Tr M := (x, u) ∈ T M | gx (u, u) = r2

g is positive definite =⇒ Tr M = Tr M. Takamichi Satoh (Tohoku Univ.)

Tangent Hyperquadric Bundles

August 31, 2010

8 / 24

Sasaki metric

Sasaki metric

Takamichi Satoh (Tohoku Univ.)

Tangent Hyperquadric Bundles

August 31, 2010

9 / 24

Sasaki metric

Splitting subspaces

p : (x, u) ∈ T M 7−→ x ∈ M : projection ∇ : Levi–Civita connection of (M, g) The tangent space (T M )(x,u) at (x, u) ∈ T M splits into the horizontal subspace H(x,u) and the vertical subspace V(x,u) with respect to ∇ : (T M )(x,u) = H(x,u) ⊕ V(x,u) .

Takamichi Satoh (Tohoku Univ.)

Tangent Hyperquadric Bundles

August 31, 2010

10 / 24

Sasaki metric

Splitting subspaces

p : (x, u) ∈ T M 7−→ x ∈ M : projection ∇ : Levi–Civita connection of (M, g) The tangent space (T M )(x,u) at (x, u) ∈ T M splits into the horizontal subspace H(x,u) and the vertical subspace V(x,u) with respect to ∇ : (T M )(x,u) = H(x,u) ⊕ V(x,u) .

Takamichi Satoh (Tohoku Univ.)

Tangent Hyperquadric Bundles

August 31, 2010

10 / 24

Sasaki metric

Lift

Let X be a vector in a tangent space Mx at x ∈ M. The horizontal lift of X to (x, u) ∈ T M is a vector X h ∈ H(x,u) s.t. p∗ X h = X. The vertical lift of X to (x, u) ∈ T M is a vector X v ∈ V(x,u) s.t. X v ( df ) = Xf for all smooth functions f on M. (Here we consider a 1-form df on M as a function on T M.)

Takamichi Satoh (Tohoku Univ.)

Tangent Hyperquadric Bundles

August 31, 2010

11 / 24

Sasaki metric

Lift

Let X be a vector in a tangent space Mx at x ∈ M. The horizontal lift of X to (x, u) ∈ T M is a vector X h ∈ H(x,u) s.t. p∗ X h = X. The vertical lift of X to (x, u) ∈ T M is a vector X v ∈ V(x,u) s.t. X v ( df ) = Xf for all smooth functions f on M. (Here we consider a 1-form df on M as a function on T M.)

Takamichi Satoh (Tohoku Univ.)

Tangent Hyperquadric Bundles

August 31, 2010

11 / 24

Sasaki metric

On tangent bundle

. . .Definition . The Sasaki metric g¯ on T M is defined at each fixed point (x, u) ∈ T M by g¯(x,u) (X h , Y h ) = gx (X, Y ), g¯(x,u) (X h , Y v ) = 0,

for ... ∀X, Y ∈ Mx .

.

.

g¯(x,u) (X v , Y v ) = gx (X, Y )

(T M )(x,u) = H(x,u) ⊕ V(x,u) , X h , Y h ∈ H(x,u) , X v , Y v ∈ V(x,u) . Takamichi Satoh (Tohoku Univ.)

Tangent Hyperquadric Bundles

August 31, 2010

12 / 24

Sasaki metric

Tangent or not

.

. . .Lemma . v The canonical vertical vector field U (x,u) := u(x,u) is perpendicular to Tr M ⊂ (T M, g¯) at each point u) ∈ T M. ..(x, . . g¯(x,u) (X h , U ) = 0,

g¯(x,u) (X v , U ) = gx (X, u).

=⇒ The horizontal lift X h is always tangent to Tr M. However, in general, the vertical lift X v is not tangent to Tr M.

Takamichi Satoh (Tohoku Univ.)

Tangent Hyperquadric Bundles

August 31, 2010

13 / 24

Sasaki metric

Tangent or not

.

. . .Lemma . v The canonical vertical vector field U (x,u) := u(x,u) is perpendicular to Tr M ⊂ (T M, g¯) at each point u) ∈ T M. ..(x, . . g¯(x,u) (X h , U ) = 0,

g¯(x,u) (X v , U ) = gx (X, u).

=⇒ The horizontal lift X h is always tangent to Tr M. However, in general, the vertical lift X v is not tangent to Tr M.

Takamichi Satoh (Tohoku Univ.)

Tangent Hyperquadric Bundles

August 31, 2010

13 / 24

Sasaki metric

Tangential lift

. .Definition . The tangential lift X t of a smooth vector field X on M is a vector field on Tr M defined by X t := X v − 

1 g¯(X v , U )U . 2 r

.

.

.

... . . . .Remark . ut = 0 for ∀(x, u) ∈ Tr M. { h }  t ⊥ (T M ) = X + Y | X ∈ M , Y ∈ {u} (⊂ M ) . x x (x,u) ... r . Takamichi Satoh (Tohoku Univ.)

Tangent Hyperquadric Bundles

August 31, 2010

14 / 24

Sasaki metric

Tangential lift

. .Definition . The tangential lift X t of a smooth vector field X on M is a vector field on Tr M defined by X t := X v − 

1 g¯(X v , U )U . 2 r

.

.

.

... . . . .Remark . ut = 0 for ∀(x, u) ∈ Tr M. { h }  t ⊥ (T M ) = X + Y | X ∈ M , Y ∈ {u} (⊂ M ) . x x (x,u) ... r . Takamichi Satoh (Tohoku Univ.)

Tangent Hyperquadric Bundles

August 31, 2010

14 / 24

Sasaki metric

On THB

=⇒ We endow the hypersurface Tr M ⊂ (T M, g¯) with the induced pseudo-Riemannian metric g˜r , which is uniquely determined by the following formulae r g˜(x,u) (X h , Y h ) = gx (X, Y ), r g˜(x,u) (X h , Y t ) = 0, r g˜(x,u) (X t , Y t ) = gx (X, Y ) − 

1 gx (X, u)gx (Y, u) r2

for ∀X, Y ∈ Mx . Takamichi Satoh (Tohoku Univ.)

Tangent Hyperquadric Bundles

August 31, 2010

15 / 24

Sasaki metric

On tangent bundle

. . .Definition . The Sasaki metric g¯ on T M is defined at each fixed point (x, u) ∈ T M by g¯(x,u) (X h , Y h ) = gx (X, Y ), g¯(x,u) (X h , Y v ) = 0,

for ... ∀X, Y ∈ Mx .

Takamichi Satoh (Tohoku Univ.)

.

Tangent Hyperquadric Bundles

August 31, 2010

.

g¯(x,u) (X v , Y v ) = gx (X, Y )

16 / 24

Proof of Theorems

Proof of Theorems

Takamichi Satoh (Tohoku Univ.)

Tangent Hyperquadric Bundles

August 31, 2010

17 / 24

Theorem 1

. 1 .Theorem . (M, g), n = dim M ≥ 3 be a pseudo-Riemannian manifold with bounded sectional curvature. =⇒ For each sufficiently small r > 0,

.

e g r )(x,u) > 0 if  = 1, Sc(˜ e g r )(x,u) < 0 if  = −1. ... Sc(˜ . e g r )(x,u) : scalar curvature of THB at (x, u) ∈ Tr M. Sc(˜

Takamichi Satoh (Tohoku Univ.)

Tangent Hyperquadric Bundles

August 31, 2010

.

Proof of Theorems

18 / 24

Proof of Theorems

positive definite

Proof of theorem 1

.

Case (i). g is positive definite. . . Theorem (Kowalski–Sekizawa, 2000) .. (M, g), dim M ≥ 3, be a Riemannian manifold with bounded sectional curvature. e g r )(x,u) > 0. =⇒ ... For each sufficiently small r > 0, Sc(˜ . The theorem 1 is the same as the theorem of Kowalski–Sekizawa.

Takamichi Satoh (Tohoku Univ.)

Tangent Hyperquadric Bundles

August 31, 2010

19 / 24

Proof of Theorems

positive definite

Proof of theorem 1

.

Case (i). g is positive definite. . . Theorem (Kowalski–Sekizawa, 2000) .. (M, g), dim M ≥ 3, be a Riemannian manifold with bounded sectional curvature. e g r )(x,u) > 0. =⇒ ... For each sufficiently small r > 0, Sc(˜ . The theorem 1 is the same as the theorem of Kowalski–Sekizawa.

Takamichi Satoh (Tohoku Univ.)

Tangent Hyperquadric Bundles

August 31, 2010

19 / 24

negative definite

Case (ii). g is negative definite. . .Lemma . Vp,q : vector space of signature (p, q)

...

.

∃φ : (x1 , . . . , xp , xp+1 , . . . , xp+q ) ∈ Vp,q 7−→ (xp+1 , . . . , xp+q , x1 , . . . , xp ) ∈ Vq,p :anti-isometry

.

.

Proof of Theorems

The theorem 1 can be proved from the lemma and the theorem of Kowalski–Sekizawa. Takamichi Satoh (Tohoku Univ.)

Tangent Hyperquadric Bundles

August 31, 2010

20 / 24

negative definite

Case (ii). g is negative definite. . .Lemma . Vp,q : vector space of signature (p, q)

...

.

∃φ : (x1 , . . . , xp , xp+1 , . . . , xp+q ) ∈ Vp,q 7−→ (xp+1 , . . . , xp+q , x1 , . . . , xp ) ∈ Vq,p :anti-isometry

.

.

Proof of Theorems

The theorem 1 can be proved from the lemma and the theorem of Kowalski–Sekizawa. Takamichi Satoh (Tohoku Univ.)

Tangent Hyperquadric Bundles

August 31, 2010

20 / 24

Proof of Theorems

indefinite

.

Case (iii). g is indefinite. . . Lemma (Kulkarni, 1979) .. (M, g) be a non-Riemannian manifold with bounded sectional curvature. =⇒ ... The sectional curvature of (M, g) is constant. . e g r ) of THB is constant. =⇒ The scalar curvature Sc(˜ e g r ) = n(n − 1)c −  n − 1 c2 r2 + (n − 1)(n − 2) 1 . Sc(˜ 2 r2 where c is the constant sectional curvature of (M, g). We proved the theorem 1. Takamichi Satoh (Tohoku Univ.)

Tangent Hyperquadric Bundles

August 31, 2010

21 / 24

Proof of Theorems

indefinite

.

Case (iii). g is indefinite. . . Lemma (Kulkarni, 1979) .. (M, g) be a non-Riemannian manifold with bounded sectional curvature. =⇒ ... The sectional curvature of (M, g) is constant. . e g r ) of THB is constant. =⇒ The scalar curvature Sc(˜ e g r ) = n(n − 1)c −  n − 1 c2 r2 + (n − 1)(n − 2) 1 . Sc(˜ 2 r2 where c is the constant sectional curvature of (M, g). We proved the theorem 1. Takamichi Satoh (Tohoku Univ.)

Tangent Hyperquadric Bundles

August 31, 2010

21 / 24

Proof of Theorems

indefinite

.

Case (iii). g is indefinite. . . Lemma (Kulkarni, 1979) .. (M, g) be a non-Riemannian manifold with bounded sectional curvature. =⇒ ... The sectional curvature of (M, g) is constant. . e g r ) of THB is constant. =⇒ The scalar curvature Sc(˜ e g r ) = n(n − 1)c −  n − 1 c2 r2 + (n − 1)(n − 2) 1 . Sc(˜ 2 r2 where c is the constant sectional curvature of (M, g). We proved the theorem 1. Takamichi Satoh (Tohoku Univ.)

Tangent Hyperquadric Bundles

August 31, 2010

21 / 24

Proof of Theorems

indefinite

.

Case (iii). g is indefinite. . . Lemma (Kulkarni, 1979) .. (M, g) be a non-Riemannian manifold with bounded sectional curvature. =⇒ ... The sectional curvature of (M, g) is constant. . e g r ) of THB is constant. =⇒ The scalar curvature Sc(˜ e g r ) = n(n − 1)c −  n − 1 c2 r2 + (n − 1)(n − 2) 1 . Sc(˜ 2 r2 where c is the constant sectional curvature of (M, g). We proved the theorem 1. Takamichi Satoh (Tohoku Univ.)

Tangent Hyperquadric Bundles

August 31, 2010

21 / 24

Proof of Theorems

indefinite

.

Case (iii). g is indefinite. . . Lemma (Kulkarni, 1979) .. (M, g) be a non-Riemannian manifold with bounded sectional curvature. =⇒ ... The sectional curvature of (M, g) is constant. . e g r ) of THB is constant. =⇒ The scalar curvature Sc(˜ e g r ) = n(n − 1)c −  n − 1 c2 r2 + (n − 1)(n − 2) 1 . Sc(˜ 2 r2 where c is the constant sectional curvature of (M, g). We proved the theorem 1. Takamichi Satoh (Tohoku Univ.)

Tangent Hyperquadric Bundles

August 31, 2010

21 / 24

Theorem 2

. 2 .Theorem . (M, g), n = dim M ≥ 2 be a pseudo-Riemannian manifold of constant sectional curvature c 6= 0. =⇒ For each sufficiently large r > 0, e g r )(x,u) < 0 if  = 1, ... Sc(˜

.

e g r )(x,u) > 0 if  = −1. Sc(˜

.

.

Proof of Theorems

Proof. e g r ) = n(n − 1)c −  n − 1 c2 r2 + (n − 1)(n − 2) 1 . Sc(˜ 2 r2

Takamichi Satoh (Tohoku Univ.)

Tangent Hyperquadric Bundles

August 31, 2010

22 / 24

Theorem 2

. 2 .Theorem . (M, g), n = dim M ≥ 2 be a pseudo-Riemannian manifold of constant sectional curvature c 6= 0. =⇒ For each sufficiently large r > 0, e g r )(x,u) < 0 if  = 1, ... Sc(˜

.

e g r )(x,u) > 0 if  = −1. Sc(˜

.

.

Proof of Theorems

Proof. e g r ) = n(n − 1)c −  n − 1 c2 r2 + (n − 1)(n − 2) 1 . Sc(˜ 2 r2

Takamichi Satoh (Tohoku Univ.)

Tangent Hyperquadric Bundles

August 31, 2010

22 / 24

Combining the thorems

. Corollary .. (M, g), n = dim M ≥ 3 be a pseudo-Riemannian manifold of constant sectional curvature c 6= 0. e g r ) = c˜. =⇒ c ∈ R, ∃r > 0 s.t. Sc(˜ ... ∀˜

.

.

.

Proof of Theorems

Proof. e g r ) = c˜. In the previous equation, we put r2 = R, Sc(˜ Then (n − 1)c2 R2 + 2(˜ c − n(n − 1)c)R − 2(n − 1)(n − 2) = 0 has a positive root. Takamichi Satoh (Tohoku Univ.)

Tangent Hyperquadric Bundles

August 31, 2010

23 / 24

Last

Thank You

Thank you for your attention ˇ ´ Dekuji vam

Takamichi Satoh (Tohoku Univ.)

Tangent Hyperquadric Bundles

August 31, 2010

24 / 24

Last

Thank You

Thank you for your attention ˇ ´ Dekuji vam

Takamichi Satoh (Tohoku Univ.)

Tangent Hyperquadric Bundles

August 31, 2010

24 / 24