The limiting case

In this section, we study the limiting case of (4.18).

Proposition 4.12 If M admits a non-zero transversal twistor spinor field Ψ of type s with the transversally conformal metric ¯gQ = e^2ugQ, i.e., P¯_X^sΨ = 0, then, for any X ∈ T M ,¯

∇_XΨ = −1

s π(X) · D_trΨ + s − q + 1

2s π(X) · d_Bu · Ψ + 1

2X(u)Ψ. (4.19) Proof. Let ¯Ψ ∈ Ker ¯P_tr^s. From (4.4), we have

∇¯_XΨ +¯ 1

sπ(X) ¯· ¯D_trΨ = 0.¯ Hence from (3.20) and (3.22), we have

∇_XΨ −s − q + 1

2s π(X) · d_Bu · Ψ −1

2X(u) ¯Ψ + 1

s π(X) · D_trΨ = 0, which yields (4.19). 2

Theorem 4.13 If (M, gM, F ) admits a non-vanishing trnsversal twistor spinor field of type q, then the foliation F is minimal.

Proof. From Theorem 4.5, the space of transversal twistor spinor space of type q is invariant under the transversally conformal metric change.

Let Φ ∈ KerP_tr^q. Then ¯Ψ = e^u2Φ ∈ Ker ¯¯ P_tr^q be the corresponding transversal twistor spinor field. Hence from (4.4), we have

∇¯E¯aΨ +¯ 1 Proof. From (2.18) and (4.20), we have

0 =X

On the other hand, from Theorem 4.13, the foliation is minimal, i.e., κ = 0. Hence from (2.18), (3.26) and (4.22), we have

D¯_tr²Ψ =¯ 1

q D¯²_trΨ +¯ 1 − q

q e^−ud_Bu ¯· ¯D_trΨ + (1 − q)e¯ ^−2u∇¯_dBuΨ +¯ 1

4σ^∇¯( ¯Ψ).

From (3.20) and (3.19), we have

∇¯_dBuΨ =∇¯ _dBuΨ −1

2 d_Bu · d_Bu · Ψ −1

2 g_Q(d_Bu, d_Bu) ¯Ψ

= − 1

q d_Bu ¯· D_trΨ + q − 1

2q |d_Bu|²Ψ¯

= − 1

q d_Bu ¯· {e^uD¯_trΨ −¯ q − 1

2 d_Bu · Ψ} + q − 1

2q |d_Bu|²Ψ¯

= − 1

q e^ud_Bu ¯· ¯D_trΨ.¯ Hence we have

D¯_tr²Ψ =¯ 1 q

D¯²_trΨ +¯ 1

4 σ^∇¯( ¯Ψ), which means (4.21). 2

Hence we have the following theorem.

Theorem 4.15 Let (M, g_M, F ) be a compact Riemannian manifold with a transverse spin foliation F of codimension q ≥ 3 and bundle-like met-ric g_M such that κ ∈ Ω¹_B(F ) and δκ = 0. Assume that K_σ^∇ ≥ 0. If there exists an eigenspinor field Φ₁ of the basic Dirac operator D_b for the eigenvalue λ₁ satisfying

λ²₁ = q

4(q − 1)(µ₁+ inf |κ|²), (4.23) then F is minimal and transversally Einsteinian with a positive constant transversal scalar curvature σ^∇.

Proof. Let D_bΦ₁ = λ₁Φ₁ with λ²₁ = _4(q−1)^q (µ₁+ inf |κ|²). From (4.8), we know that ¯Ψ₁ ∈ Ker ¯P_tr^q with Ψ₁ = e^{−q−12 u}Φ₁. Since ¯D_bΨ¯₁ = λ₁e^−uΨ¯₁,

we have

D¯²_bΨ¯₁ = −λ₁e^−2ud_Bu · Ψ₁+ λ²₁e^−2uΨ¯₁.

From (4.21), we have

λ²₁Ψ¯₁ = q

4(q − 1) e^2uσ^∇¯Ψ¯₁+ λ₁d_Bu · Ψ₁. Hence we get

λ²₁| ¯Ψ₁|² = q

4(q − 1) e^2uσ^∇¯| ¯Ψ₁|² and d_Bu · Ψ₁ = 0, (4.24) which means u is constant and e^2uσ^∇¯ is positive and constant. So Propo-sition 4.12 implies that for ¯Ψ₁ ∈ Ker ¯P_tr^q, Ψ₁ = e^{−q−12 u}Φ₁,

∇_XΦ₁ = −λ₁

q X · Φ₁. (4.25)

By direct calculation with (4.25), we have

∇_X∇_EaΦ₁ =∇_X{−λ₁

q E_a· Φ₁}

= −λ₁

q (∇_XE_a) · Φ₁− λ₁

q E_a· ∇_XΦ₁

= −λ₁

q (∇_XE_a) · Φ₁+ λ²₁

q² E_a· X · Φ₁ for any X ∈ ΓQ. Hence we have

R^S(X, E_a) · Φ₁ =2λ²₁

q² E_a· X · Φ₁ +2λ²₁

q² g_Q(X, E_a)Φ₁ and so we have

X

a

E_a· R^S(X, E_a) · Φ₁ = −2λ²₁

q² (q − 1)X · Φ₁. (4.26) On the other hand, from (2.21) and (4.24), we have

λ²₁ = q

4(q − 1) σ^∇. (4.27)

Therefore we have, from (3.12), (4.26), and (4.27), ρ^∇(X) =σ^∇

q X. (4.28)

This means that F is transversally Einsteinian with a positive constant transversal scalar curvature σ^∇. 2

5 Eigenvalue estimate with a modified con-nection

5.1 Eigenvalue estimate

Let (M, g_M, F , S(F )) be a Riemannian manifold with a transverse spin foliation F of codimension q and a bundle-like metric g_M such that κ is basic-harmonic.

Now, we introduce a new connection ^f,g∇ on S(F ) as the following:

Definition 5.1 Let f and g be real-valued basic functions on M . For any tangent vector field X and any spinor field Ψ, we define the modified connection ^f,g∇ on S(F ) by

f,g

∇X Ψ = ∇_XΨ + f π(X) · Ψ + gκ · π(X) · Ψ, (5.1) where π : T M → Q.

Lemma 5.2 Let (M, gM, F ) be a Riemannian manifold with a trans-verse spin foliation F and a bundle-like metric g_M. Then, for any basic-harmonic 1-form ω ∈ Ω¹_B(F ),

D_tr(ω · Ψ) = −ω · D_trΨ − 2∇_ωΨ. (5.2)

Proof. For any spinor field Ψ, a simple calculation gives D_tr(ω · Ψ) =X

a

E_a· (∇_Eaω) · Ψ +X

a

E_a· ω · ∇_EaΨ −1

2 κ · ω · Ψ

=X

a

{Ea∧ ∇Eaω − i(Ea)∇Eaω}Ψ −X

a

ω · Ea· ∇EaΨ

− 2X

a

gQ(Ea, ω)∇EaΨ −1

2 {−ω · κ · Ψ − 2gQ(κ, ω)Ψ}

=(d_Bω + δ_Bω − i(κ_B)ω)Ψ −X

a

ω · E_a· ∇_EaΨ − 2∇_ωΨ +1

2 ω · κ · Ψ + g_Q(κ, ω)Ψ

= − ω · D_trΨ − 2∇_ωΨ + (d_Bω + δ_Bω)Ψ.

Since ω ∈ Ω¹_B(F ) is a basic-harmonic 1-form, we have dBω = 0 = δBω.

Hence the proof is complete. 2

Proposition 5.3 For any real-valued basic functions f and g on M , and for any spinor field Ψ ∈ S(F ), we have

|^f,g∇tr Ψ|² = |∇_trΨ|²+ qf²|Ψ|²+ qg²|κ|²|Ψ|²+ g|κ|²|Ψ|² (5.3)

− 2f Re < D_trΨ, Ψ > +2gRe < D_trΨ, κ · Ψ >

− 4gRe < ∇_κΨ, Ψ > .

Proof. Fix x ∈ M and choose an orthonormal basic frame {E_a} such that (∇E_a)_x = 0 for all a. Then we have at the point x that for any Ψ,

|^f,g∇tr Ψ|² =X

a

<^f,g∇Ea Ψ,^f,g∇Ea Ψ >

= |∇_trΨ|²+ qf²|Ψ|²+ qg²|κ|²|Ψ|²+ g|κ|²|Ψ|²

− f {< D_trΨ, Ψ > + < Ψ, D_trΨ >}

+ g{< D_trΨ, κ · Ψ > + < κ · Ψ, D_trΨ >}

− f Re < Ψ, κ · Ψ > −4gRe < ∇_κΨ, Ψ >,

which means (5.3) together with the fact that < X · Ψ, Ψ > is pure imaginary. 2

We now that for an appropriate choice of the real-valued basic func-tions f and g, one gets a sharp estimate of the first eigenvalue of the basic Dirac operator on compact foliated Riemannian manifolds.

Theorem 5.4 Let (M, g_M, F ) be a Riemannian manifold with an isopara-metric transverse spin foliation of codimension q > 1 and bundle-like metric g_M with respect to F . Assume that the mean curvature κ of F satisfies δ_Bκ = 0 and K_σ^∇ ≥ 0. Any eigenvalue λ of the transverse Dirac

Hence from Proposition 5.3, we have Z

Corollary 5.5 In addition to assumptions in Theorem 5.4, if the trans-verse scalar curvature is zero, then we get

λ² ≥ q + 1 4(q − 1) inf

M |κ|².

5.2 The limiting case

We define Ric^f,g_∇ : ΓQ ⊗ S → S by Ric^f,g_∇ (X ⊗ Ψ) =X

a

E_a· R^f,g(X, E_a)Ψ, (5.6)

where R^f,g is the curvature tensor with respect to ^f,g∇. Then we have the following lemma.

Lemma 5.6 For any vector field X ∈ ΓQ and spinor field Ψ ∈ ΓS(F ),

Ric^f,g_∇ (X ⊗ Ψ) (5.7)

= −1

2 ρ^∇(X)Ψ − qX(f )Ψ + 2(q − 1)f²X · Ψ − dBf · X · Ψ

+ (q − 2)X(g)κ · Ψ + (q − 2)g∇Xκ · Ψ + 2qf ggQ(X, κ)Ψ + 2f gκ · X · Ψ + 2(q − 2)g²|κ|²X · Ψ − 2(q − 2)g²gQ(X, κ)κ · Ψ − dBg · κ · X · Ψ + g|κ|²X · Ψ.

Proof. A direct calculation gives

f,g

∇X f,g

∇Ea Ψ =^f,g∇X {∇_EaΨ + f E_a· Ψ + gκ · E_a· Ψ}

= ∇_X∇_EaΨ + X(f )E_a· Ψ + f ∇_XE_a· Ψ + f E_a· ∇_XΨ + X(g)κ · E_a· Ψ + g∇_Xκ · E_a· Ψ + gκ · ∇_XE_a· Ψ + gκ · E_a· ∇_XΨ + f X · ∇_EaΨ + f²X · E_a· Ψ

+ f gX · κ · E_a· Ψ + gκ · X · ∇_EaΨ + f gκ · X · E_a· Ψ + g²κ · X · κ · E_a· Ψ.

With the similar calculation, we have

R^f,g(X, Ea)Ψ = R^S(X, Ea)Ψ + X(f )Ea· Ψ − X(g)Ea· κ · Ψ

− 2X(g)gQ(κ, Ea)Ψ − gEa· ∇Xκ · Ψ − 2ggQ(∇Xκ, Ea)Ψ

− 2f²Ea· X · Ψ − 2f²gQ(X, Ea)Ψ

− 2f ggQ(X, κ)Ea· Ψ + 2f ggQ(Ea, κ)X · Ψ

+ g²κ · {X · κ · E_a− E_a· κ · X} · Ψ − E_a(f )X · Ψ

− E_a(g)κ · X · Ψ − g∇_Eaκ · X · Ψ.

Note that

X · κ · E_a− E_a· κ · X = 2κ · E_a· X + 2g_Q(X, E_a)κ − 2g_Q(X, κ)E_a + 2g_Q(E_a, κ)X.

Hence we have

R^f,g(X, E_a)Ψ = R^S(X, E_a)Ψ + X(f )E_a· Ψ − X(g)E_a· κ · Ψ

− 2X(g)g_Q(κ, E_a)Ψ − gE_a· ∇_Xκ · Ψ − 2gg_Q(∇_Xκ, E_a)Ψ

− 2f²E_a· X · Ψ − 2f²g_Q(X, E_a)Ψ − 2f gg_Q(X, κ)E_a· Ψ + 2f gg_Q(E_a, κ)X · Ψ − 2g²|κ|²E_aX · Ψ

− 2g²g_Q(X, E_a)|κ|²Ψ + 2g²g_Q(X, κ)E_a· κ · Ψ + 4g²g_Q(X, κ)g_Q(E_a, κ)Ψ + 2g²g_Q(E_a, κ)κ · X · Ψ

− E_a(f )X · Ψ − E_a(g)κ · X · Ψ − g∇_Eaκ · X · Ψ.

From (3.12) and (5.6), the proof is completed. 2 Let D_bΨ₁ = λ₁Ψ₁ with λ²₁ = q

4(q − 1) inf

M



K_σ^∇+1 q|κ|²

. From (5.5), we see ^f1∇^,g1tr Ψ1 = 0, where f1 = λ₁

q and g1 = − 1

2q. Hence from (5.1), we have

∇XΨ1 = −λ₁

q X · Ψ1+ 1

2q κ · X · Ψ1. (5.8)

Note that

X

a

E_a· ∇_EaΨ₁ =λ₁Ψ₁+ 1

2 κ · Ψ₁. On the other hand, from (5.8)

X the left-hand side of (5.10) is pure imaginary. But the right-hand side of (5.10) is real. Therefore, both sides are zero. Hence, if q ≥ 2, then we have dBf = 0. That is, X(f ) = 0 for any X ∈ ΓQ. Since f is basic function, f is constant. So from (5.9), we have

ρ^∇(X)Ψ = 4(q − 1)

q² λ²₁X · Ψ. (5.11) This means that F is transversally Einsteinian with a constant transver-sal scalar curvature σ^∇= 4(q − 1)

q λ²₁. Hence we have the following the-orem.

Theorem 5.7 Let (M, g_M, F ) be a compact Riemannian manifold with a transverse spin foliation F of codimension q > 1 and a bundle-like

metric g_M. Assume that K_σ^∇≥ 0. If there exists an eigenspinor field Ψ₁ of the basic Dirac operator D_b for the eigenvalue λ₁ satisfying

λ²₁ = q

4(q − 1) inf

M



K_σ^∇+ 1 q |κ|²

, (5.12)

then F is minimal and transversally Einsteinian with a positive constant transversal scalar curvature σ^∇.

6 Eigenvalue estimate with the conformal change

6.1 Eigenvalue estimate

Let (M, g_M, F , S(F )) be a Riemannian manifold with a transverse spin foliation F of codimension q and a bundle-like metric g_M such that κ is basic-harmonic. In this section, we estimate the eigenvalues of the basic Dirac operator by a transversally conformal change of the metric.

Now, we consider, for any real basic function u on M , the transversally conformal metric ¯g_Q = e^2ug_Q. Let ¯S(F ) be its corresponding spinor bundle. For any tangent vector field X and any spinor field Ψ, we define the modified connection

f,g∇ on ¯¯ S(F ) by

f,g∇¯_X Ψ = ¯¯ ∇_XΨ + f π(X) ¯· ¯¯ Ψ + gκ_¯g¯· π(X) ¯· ¯Ψ, (6.1)

where f and g are real-valued basic functions on M .

Lemma 6.1 Let (M, g_M, F ) be a Riemannian manifold with a trans-verse spin foliation F and a bundle-like metric g_M. Then for any

basic-harmonic 1-form ω ∈ Ω¹_B(F ),

D¯_tr(f ω ¯· ¯Ψ) = − f ω ¯· ¯D_trΨ − 2f ¯¯ ∇_ωΨ − (q + 2)f ω(u) ¯¯ Ψ (6.2)

− 2f ω · d_Bu · Ψ + d_Bf · ω · Ψ where f is any basic function.

Proof. Note that we have, from (3.23),

D_tr(f ω · Ψ) =d_Bf · ω · Ψ + f D_tr(ω · Ψ)

= − f ω · D_trΨ − 2f ∇_ωΨ + d_Bf · ω · Ψ.

From (3.20), (3.22) and (3.24), we have D¯_tr(f ω ¯· ¯Ψ)

= ¯D_tr(e^uf ω · Ψ) = e^−ud_Be^u· f ω · Ψ + e^uD¯_tr(f ω · Ψ)

=f dBu · ω · Ψ + Dtr(f ω · Ψ) + q − 1

2 dBu · f ω · Ψ

= − f ¯ω ¯· DtrΨ − 2f ∇ωΨ + dBf · ω · Ψ + q + 1

2 f dBu · ω · Ψ

= − f ¯ω ¯·



e^uD¯trΨ −¯ q − 1

2 dBu · Ψ



− 2f ¯∇_ωΨ +¯ 1

2 ω · d_Bu · Ψ + 1

2 ω(u) ¯Ψ + d_Bf · ω · Ψ − q + 1

2 f ω · d_Bu · Ψ − (q + 1)f ω(u) ¯Ψ

= − f ω ¯· ¯D_trΨ +¯ q − 1

2 f ω · d_Bu · Ψ − 2f ¯∇_ωΨ¯

− f ω · d_Bu · Ψ − f ω(u) ¯Ψ + d_Bf · ω · Ψ

−q + 1

2 f ω · d_Bu · Ψ − (q + 1)f ω(u) ¯Ψ

= − f ω ¯· ¯D_trΨ − 2f ¯¯ ∇_ωΨ¯

− 2f ω · d_Bu · Ψ − (q + 2)ω(u) ¯Ψ + d_Bf · ω · Ψ, which implies (6.2). 2

Let K_u = {u ∈ Ω⁰_B(F ) | κ(u) = 0}. Then we have the following

Proposition 6.3 For any real-valued basic functions f and g on M , and for any spinor field Ψ, we have

| Proof. This is a simple calculation. 2

Let D_bΦ = λΦ (Φ 6= 0) and ¯Ψ = e^{−q+12 u}Φ. Since < X · Ψ, Ψ > is pure

From (6.3) and (3.30), if u ∈ K_u, then Hence we have the following theorem.

Theorem 6.4 Let (M, g_M, F ) be a compact manifold with a transverse spin foliation F of codimension q ≥ 2 and bundle-like metric g_M such that κ ∈ Ω¹_B(F ) and δκ = 0. Assume that K_σ^∇¯ ≥ 0 for some transversally

From (4.16), we have the following corollary.

Corollary 6.5 Under the same condition as in Corollary 4.10, we have

λ² ≥

Corollary 6.6 Let (M, g_M, F ) be a compact Riemannian manifold with a transverse spin foliation F of codimension q ≥ 3 and bundle-like metric g_M with κ ∈ Ω¹_B(F ) and δκ = 0. If the transversal scalar curvature sat-isfies σ^∇≥ 0, then any eigenvalue λ of the Dirac operator corresponding to the eigenspinor Ψ satisfies

where µ₁ is the first eigenvalue of the basic Yamabe operator Y_b of F .

6.2 The limiting case

We define Ric^f,g_∇¯ : ΓQ ⊗ ¯S(F ) → ¯S(F ) by Ric^f,g_∇¯ (X ⊗ ¯Ψ) =X

a

E¯_a¯· ¯R^f,g(X, ¯E_a) ¯Ψ, (6.9)

where ¯R^f,g is the curvature tensor with respect to

f,g∇. For X ∈ ΓQ and¯ Ψ ∈ ΓS(F ) we have the following by the direct calculation;

f,g

∇¯_X

f,g

∇¯E¯a Ψ =¯

f,g

∇¯_X  ¯∇E¯aΨ + f ¯¯ E_a¯· ¯Ψ + gκ_g¯¯· ¯E_a¯· ¯Ψ

= ¯∇_X∇¯E¯aΨ + f X ¯· ¯¯ ∇E¯aΨ + gκ¯ _g¯¯· X ¯· ¯∇E¯aΨ¯ + X(f ) ¯E_a¯· ¯Ψ + f ¯∇_XE¯_a¯· ¯Ψ + f ¯E_a¯· ¯∇_XΨ¯ + f²X ¯· ¯Ea¯· ¯Ψ + f gκg¯¯· X ¯· ¯Ea¯· ¯Ψ

+ X(g)κg¯¯· ¯Ea¯· ¯Ψ + g ¯∇Xκ¯g¯· ¯Ea¯· ¯Ψ + gκ¯g¯· ¯∇XE¯a¯· ¯Ψ + gκ¯g¯· ¯Ea¯· ¯∇XΨ + f gX ¯· κ¯ ¯g¯· ¯Ea¯· ¯Ψ

+ g²κ¯g¯· X ¯· κ¯g¯· ¯Ea¯· ¯Ψ.

With the similar calculation, we have R¯^f,g(X, ¯E_a) ¯Ψ

= ¯R^S(X, ¯E_a) ¯Ψ + X(f ) ¯E_a¯· ¯Ψ − 2f²E¯_a¯· X ¯· ¯Ψ − 2f²g¯_Q(X, ¯E_a) ¯Ψ

− 2f g¯g_Q(κ_¯g, X) ¯E_a¯· ¯Ψ − X(g) ¯E_a¯· κ_g¯¯· ¯Ψ

− 2X(g)¯g_Q(κ_¯g, ¯E_a) ¯Ψ − g ¯E_a¯· ¯∇_Xκ_g¯¯· ¯Ψ − 2g¯g_Q( ¯∇_Xκ_¯g, ¯E_a) ¯Ψ − ¯E_a(f )X ¯· ¯Ψ + g²κ_g¯¯· 

X ¯· κ_¯g¯· ¯E_a− ¯E_a¯· κ_g¯¯· X

¯· ¯Ψ + 2f g¯g_Q(κ_g¯, ¯E_a)X ¯· ¯Ψ

− ¯E_a(g)κ_¯g¯· X ¯· ¯Ψ − g ¯∇E¯aκ_¯g¯· X ¯· ¯Ψ.

Note that

X ¯· κ_¯g¯· ¯E_a− ¯E_a¯· κ_g¯¯· X = 2κ_¯g¯· ¯E_a¯· X + 2¯g_Q(X, ¯E_a)κ_¯g

− 2¯g_Q(X, κ_¯g) ¯E_a+ 2¯g_Q(κ_¯g, ¯E_a)X.

Hence we have R¯^f,g(X, ¯E_a) ¯Ψ

= ¯R^S(X, ¯E_a) ¯Ψ + X(f ) ¯E_a¯· ¯Ψ − 2f²E¯_a¯· X ¯· ¯Ψ − 2f²g¯_Q(X, ¯E_a) ¯Ψ

− 2f g¯gQ(κ¯g, X) ¯Ea¯· ¯Ψ − X(g) ¯Ea¯· κ¯g¯· ¯Ψ − ¯Ea(f )X ¯· ¯Ψ

− 2X(g)¯gQ(κ¯g, ¯Ea) ¯Ψ − g ¯Ea¯· ¯∇Xκg¯¯· ¯Ψ − 2g¯gQ( ¯∇Xκ¯g, ¯Ea) ¯Ψ

− 2g²|κ¯g|²E¯a¯· X ¯· ¯Ψ − 2g²|κg¯|²¯gQ(X, ¯Ea) ¯Ψ + 2g²g¯Q(X, κ¯g) ¯Ea¯· κg¯¯· ¯Ψ + 4g²g¯Q(X, κg¯)¯gQ(κg¯, ¯Ea) ¯Ψ + 2g²¯gQ(κ¯g, ¯Ea)κg¯¯· X ¯· ¯Ψ

+ 2f g¯gQ(κ¯g, ¯Ea)X ¯· ¯Ψ − ¯Ea(g)κ¯g¯· X ¯· ¯Ψ − g ¯∇E¯aκ¯g¯· X ¯· ¯Ψ.

By a simple calculation, we have, from (3.12) and (6.9),

Ric^f,g_∇¯ (X ⊗ ¯Ψ) (6.10)

= −1

2 ρ^∇¯(X) ¯· ¯Ψ − qX(f ) ¯Ψ + 2(q − 1)f²X ¯· ¯Ψ + 2qf g¯g_Q(κ_¯g, X) ¯Ψ + (q − 2)X(g)κ_¯g¯· ¯Ψ + (q − 2)g ¯∇_Xκ_g¯¯· ¯Ψ

− d_Bf ¯· X ¯· ¯Ψ + 2(q − 2)g²|κ_¯g|²X ¯· ¯Ψ − 2(q − 2)g²g¯_Q(X, κ_¯g)κ_g¯¯· ¯Ψ

− 2f gκ_g¯¯· X ¯· ¯Ψ − d_Bg ¯· κ_g¯¯· X ¯· ¯Ψ + g|κ_g¯|²X ¯· ¯Ψ.

On the other hand, we have the following.

Proposition 6.7 If a non-zero spinor field Ψ satisfies

f,g∇¯_tr Ψ = 0, then¯

∇_XΨ = −f e^uπ(X) · Ψ − gκ · π(X) · Ψ (6.11) + 1

2 g_Q(d_Bu, π(X))Ψ + 1

2 π(X) · d_Bu · Ψ.

Proof. From (6.1), we have

∇¯_XΨ + f π(X) ¯· ¯¯ Ψ + gκ_¯g¯· π(X) ¯· ¯Ψ = 0.

Hence from (3.20), we have

∇_XΨ −1

2 π(X) · d_Bu · Ψ −1

2X(u) ¯Ψ + f e^uπ(X) · Ψ + gκ · π(X) · Ψ = 0.

Since ˜I_u is an isometry, the proof is completed. 2

Theorem 6.8 Let (M, g_M, F ) be a compact Riemannian manifold with a transverse spin foliation F of codimension q ≥ 3 and bundle-like metric g_M such that κ ∈ Ω¹_B(F ) and δκ = 0. Assume that σ^∇≥ 0. If there exists an eigenspinor field Ψ₁ of the basic Dirac operator D_b for the eigenvalue λ²₁ = q

4(q − 1)



µ₁ + q + 1

q inf |κ|²

, then F is minimal, transversally Einsteinian with a positive constant transversal scalar curvature σ^∇. Proof. Let D_bΦ = λ₁Φ with λ²₁ = q

Hence the left hand side in the equation (6.12) is pure imaginary but the right hand side in the equation (6.12) is real, and so both sides are all zero. That is, d_Bf = 0. So u is constnat. Also, we have from (6.10)

ρ^∇¯(X) = 4(q − 1)f²X for X ∈ ΓQ. (6.13) Since u is constant, we have from (2.20)

ρ^∇(X) = 4(q − 1)

q² λ²₁X. (6.14)

Hence F is transversally Einsteinian with a constant transversal scalar curvature σ^∇ = 4(q − 1)

q λ²₁. 2

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