On the Killing Tensor Fields on a Compact Riemannian Manifold

Grigorios Tsagas

Abstract

Let (M, g) be a compact Riemannian manifold of dimension n. The aim of the present paper is to study the dimension of K^{q}(M, R) in the connection with the Riemannian metric g on M.

Mathematics Subject Classification: 53C20 Key words: Riemannian manifold, Killing tensor field, Riemannian metric, harmonic q-form and Killing q-form.

1

Let (M, g) be a compact Riemannian manifold of dimension n. Let K^{q}(M, R) where q = 2, ..., n − 1 be the vector space of Killing tensor fields of order q on M. The study of the dimension of K^{q}(M, R) is an important problem. This importance comes from the fact there is a connection between q-harmonic forms and Killing tensor fields of order q. Let H^{q}(M, R) be the vector space of harmonic q-forms. It is known that dim(H^{q}(M, R)) = b_{q} is the q-Betti number of M, which is topological invariant. It is still open if dim(K^{q}(M, R)) for q = 2, ..., n − 1 is also a topological invariant.

The aim of the present paper is to study this problem. We also improve Yano’s results ([16]).

The whole paper contains three paragraphs. Each of them is analyzed as follows.

In the second paragraph we study differential operators of cross sections of a fibre bundle over a compact Riemannian manifold M. The Killing tensor fields of order q can be considered as special cross sections of the fibre bundle ∇^{q}(T (M)) over M.

The space of Killing tensor fields K^{q}(M, R) of order q with the connection of the Riemannian metric g on M is studied in the last paragraph. These results are an improvement Yano’s results ([16]).

Balkan Journal of Geometry and Its Applications, Vol.1, No.2, 1996, pp. 91-97 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

2

Let (M, g) be a compact Riemannian manifold of dimension n without boundary. We denote by ∧^{q}(T (M)) and ∧^{q}(T^{⋆}M) the fibre bundles of antisymmetric convariant tensor fields of order q and antisymmetric contravariant tensor fields of order q respectively on the manifold M. It is known that the vector space ∧^{q}(T^{⋆}M) coincide with the vector space ∧^{q}(M) of exterior q-forms.

We must notice that each exterior q-form w is a cross section of ∧^{q}(T^{⋆}M) = ∧^{q}(M). The same is true for each element λ ∈ ∧^{q}(T M). The Laplace operator Δ is a second order elliptic differential operator C^{∞}(∧^{q}(M)), that is

Δ = dδ + δd : C^{∞}(∧^{q}(M)) → C^{∞}(∧^{q}(M)),
Δ = dδ + δd : α → Δ(α) = dδ(α) + δd(α), α ∈ C^{∞}(∧^{q}(M)),

where α an exterior q-form and d, δ are first order differential operator defined by

d : C^{∞}(∧^{q}(M)) → C^{∞}(∧^{q+1}(M)),
δ : C^{∞}(∧^{q}(M)) → C^{∞}(∧^{q−1}(M)).

These differential operators are related by

< α, δβ >=< dα, β >, ∀α ∈ C^{∞}(∧^{q}(M)), ∀β ∈ C^{∞}(∧^{q+1}(M)),

where <> is the global inner product on C^{∞}(∧^{q}(M)). The local inner product is defined by

< α, γ >_{1}= α_{i_{1},...,i_{q}} β^{i_{1},...,i_{q}} = g^{j_{1}i_{1}...}g^{j_{q}i_{q}} α_{i_{1},...,i_{q}} β_{j_{1},...,j_{q}}.

Let (x_{1}, ..., x_{n}) be a local coordinate system on the chart (U, φ) and let {e_{1}, ..., e_{n}} be the associated local frame in M, that is

e_{1} = \frac{∂}{∂x_{1}} , ..., e_{n} = \frac{∂}{∂x_{n}} .

If α is a q-form, which is a cross section of ∧^{q}(M), that is α ∈ C^{∞}(∧^{q}(M)), then α with respect to the local coordinate system can be expressed by

α(e_{i_{1}}, e_{i_{2}}, ..., e_{i_{q}}) = α_{i_{1},...,i_{q}}, 1 ≤ i_{1} < i_{2} < ... < i_{q} ≤ n.

The following formulas are known

(2.1)  (dα)_{i_{1}...i_{q}j} = \frac{1}{q!}ε_{i_{1}...i_{q}j}^{kj_{1}...j_{q}}∇_{k}α_{j_{1}...j_{q}j}

(2.2)  (δα)_{i_{2}...i_{q}} = −∇_{l}α_{i_{2}...i_{q}}^{l}
(Δα)_{i_{1}...i_{q}} = −∇^{k}∇_{k}α_{i_{1}...i_{q}}+

(2.3)  +\frac{1}{(q − 1)!} ε_{i_{1}...i_{q}}^{kj_{2}...j_{q}}(∇_{l}∇_{k}α_{i_{2}...i_{q}}^{l} − ∇_{k}∇_{l}α_{i_{2}...i_{q}}^{l}),

where

ε_{j_{1}...j_{r}}^{i_{1}...i_{r}} = \{ \begin{matrix}1 & if (i_{1}...i_{r}) is even permutation of (j_{1}...j_{r}) \\ −1 & if (i_{1}...j_{r}) is odd permutation of (j_{1}...j_{r}). \\ 0 & if (i_{1}...i_{r}) is not permutation of (j_{1}...j_{r}) \end{matrix}

The formula (2.3), by means of Ricci’s formula, becomes

∇_{l}∇_{k}α_{i_{2}...i_{q}}^{l} − ∇_{k}∇_{l}α_{i_{2}...i_{q}}^{l} = R_{rlk}^{l}α_{i_{2}...i_{q}}^{r}−

(2.4)  − ∑_{s=2}^{q} R_{i_{s}lk}^{r}α_{i_{2}...i_{s−1}ri_{s+1}...i_{q}}^{l},

and after some estimates, takes the form

(Δα)_{i_{1}...i_{q}} = −∇^{k}∇_{k}α_{i_{1}...i_{q}} + \frac{1}{(q − 1)!} ε_{i_{1}...i_{q}}^{kj_{2}...j_{q}}R_{kl}α_{j_{2}...j_{q}}^{l}−

(2.5)  −\frac{1}{2(q − 2)!} ε_{i_{1}...i_{q}}^{klj_{3}...j_{q}}R_{klmn}α_{j_{2}...j_{q}}^{mn}.

If α is a q-form, then we have

(2.6)  \frac{1}{2}Δ(| α |^{2}) = (α, Δα)− | ∇α |^{2} −\frac{1}{(q − 1)!}L_{q}(α),

where q ≥ 2 and

(2.7)  | ∇α |^{2}= \frac{1}{q!}∇^{k}α^{i_{1}...i_{q}} ∇_{k}α_{i_{1}...i_{q}},

(2.8)  L_{q}(α) = −(q − 1)R_{klmn}α^{kli_{3}...i_{q}} α_{j_{3}...i_{q}}^{mn} + 2R_{kl}α^{ki_{2}...i_{q}}α_{i_{2}...i_{q}}^{l}.

From (2.8) we can consider L_{q} as a quadratic form on the vector space ∧^{q}(M, R), that is

(2.9)  L_{q} : ∧^{q}(M, R) → R, L_{q} : α → L_{q}(α).

A q-form α is called killing q-form if its covariant derivative ∇ α is a (q + 1)- form. This in local system (x_{1}, ..., x_{n}) can be expressed as follows

(2.10)  ∇_{j}α_{ii_{2}...i_{q}} + ∇_{i}α_{ji_{2}...i_{q}} = 0,

which is equivalent to

(2.11)  q∇_{j}α_{i_{1}i_{2}...i_{q}} + ∇_{i_{1}}α_{ji_{2}...i_{q}} + ... + ∇_{i_{q}}α_{i_{1}i_{2}...i_{q−1}j} = 0.

If α is a killing q-form, then from (2.11) we obtain

(2.12)  ∇_{j}α_{i_{2}...i_{q}}^{j} = 0.

The killing q-form α satisfies the equations

qg^{jk}∇_{k}∇_{j} α_{i_{1}...i_{q}} + ∑_{s}^{1...q} α_{i_{1}...i_{s−1}}r i_{s+1}...i_{q}R_{i_{s}}^{r}+

(2.12)  + ∑_{s<t}^{1...q} α_{i_{1}...i_{s−1}}r i_{s+1}...i_{t−1} μ i_{t+1}...i_{q} R^{rμ} i_{s} i_{t} = 0

Hence if we consider the second order elliptic differential operator

D_{q} : C^{∞}(∧^{q}(M, R)) → C^{∞}(∧^{q}(M, R))
D_{q} : α → D_{q} α,

where

(D_{q} α)_{i_{1}...i_{q}} = q g^{jk}∇_{k}∇_{j} α_{i_{1}...i_{q}} + ∑_{s}^{1...q} α_{i_{1}...i_{s−1} r i_{s+1}...i_{q}} R_{i_{s}}^{r}+

(2.13)  + ∑_{s<t}^{1...q} α_{i_{1}...i_{s−1} r i_{r+1}...i_{t−1} μ i_{t+1}...i_{q}}

Therefore the Ker(D_{q}) of D_{q}, that is

Ker(D_{q}) = {α ∈ ∧^{q}(M, R)/D_{q}(α) = 0}

consists of the killing q-forms, whose space is denoted by K_{q}(M, R), that means K_{q}(M, R) = Ker(D_{q}). Proposition 2.1.. There is an isomorphism between the vector spaces AD_{q}(M, R) and AD^{q}(M, R), where AD_{q}(M, R) and AD^{q}(M, R) are the vector spaces of antisymmetric convariant tensor fields of order q, that is q-forms, and antisymmetric contravariant tensor fields of order q respectively. Proof. Let (U, ϕ) be a chart of M with local coordinate system (x_{1}, ..., x_{n}). If w is a q-form on M, then w has the following components

{w_{i_{1}...i_{q}}/1 ≤ i_{1} < i_{2} < ... < i_{q} ≤ n},

with respect to the local coordinate system (x_{1}, ..., x_{n}). We consider the following linear mapping

F : AD_{q}(M, R) = ∧^{q}(M, R) → AD^{q}(M, R)
F : w → F (w)

whose component of F (w) with respect to (x_{1}, ..., x_{n}) are the following

F (w)^{i_{1}...j_{q}} = g^{i_{1}j_{1}}...g^{i_{q}j_{q}} w_{i_{1}...i_{q}}.

It can be easily proved that F is bijective. Therefore the vector spaces AD_{q}(M, R) and AD^{q}(M, R) are isomorphic, q.e.d. Remark 2.2. If w is a killing q-form, then F (w), which is an antisymmetric contravariant tensor field of order q, has the property ∇F (w) = 0. An antisymmetric contravariant tensor field β of order q with the property ∇β = 0, is called killing tensor field of order q. Due to isomorphism F we can use the notion killing tensor field of order q instead of killing q-form and conversely.

3

The set of killing tensor fields of order q is denoted by K^{q}(M, R), which is isomorphic onto K_{q}(M, R).

In this paragraph we shall study the dim(K^{q}(M, R)) with respect to some properties of the Riemannian metric g on M.

If α is a killing q-form, then

(3.1)  (α, Δα) − (Δα)_{i_{1}...i_{q}}α^{i_{1}...i_{q}},

which by means of (2.5) and after some estimates and taking under to consideration (2.8) we obtain

(3.2)  (α, Δα) = −\frac{(q + 1)}{q!}L_{q}(α).

The equation (2.6) by means of (3.2) becomes

(3.3)  \frac{1}{2}Δ(| α |^{2}) = − | ∇α |^{2} +\frac{(q + 1)}{q!}L_{q}(α).

From the second order elliptic differential operator D_{q} we obtain and endomorphism (D_{q})_{x} of the fibre ∧^{q}(M, R)_{x} in x, that is

(3.4)  (D_{q})_{x} : ∧^{q}(M, R)_{x} → ∧^{q}(M, R)_{x},

which satisfies the relation

< (D_{q})_{x} u, v >=< u, (D_{q})_{x}v >, ∀u, v ∈ ∧^{q}(M, R),

where <> is the inner product on ∧^{q}(M, R)_{x} induced by the inner product on T^{⋆}M.

Now, we define

(3.5)  R(x) = Sup {< (D_{q})_{x}v, v > /v ∈ ∧^{q}(M, R), < v, v >= 1}

(3.6).  R_{max} = Sup{R(x)/x ∈ M}

Now, we shall prove the theorem Theorem 3.1. Let (M, g) be a compact Riemannian manifold of dimension n. If R(x) ≤ 0 and there exists an x_{0} such that R(x_{0}) < 0, then K^{q}(M, R) = {0}.

If R_{max} = 0, then dimK^{q}(M, R) ≤ 1 = rank{∧^{q}(M, R)}. Proof. If we integrate (3.3) on the manifold M, we obtain

(3.7)  ∫_{M} [− | ∇α |^{2} +\frac{q + 2}{2q!}L_{q}(α)] dM = 0.

From the inequalities

(3.8)  − | ∇α |^{2}≤ 0

and the assumptions that R(x) ≤ 0, ∀x ∈ M − {x_{0}} and R(x_{0}) < 0, which imply

(3.9)  L_{q}(x) ≤ 0, ∀x ∈ M − {x_{0}} and L_{q}(x_{0}) < 0,

we conclude that

(3.10)  ∇α = 0 and α/x = 0, ∀x ∈ M,

which yields

α = 0.

This proves that K^{q}(M, R) = {0}

If R_{max} = 0, then the formula (3.7) implies

(3.11)  ∫_{M}[− | ∇α |^{2}]dm + \frac{q + 2}{q!} ∫_{M} L_{q}(α)dM ≤ 0

which implies | ∇α |= 0, that means α is a parallel tensor field. Hence every killing tensor field of order q on M is parallel. Since the maximal number of independent parallel killing tensor fields on M is less or equal than the rank(E), where E is the vector bundle of exterior q-forms, then we have

dim(K^{q}(M, R) ≤ 1 = rank{∧^{q}(M, R)} q.e.d.

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Aristotle University of Thessaloniki School of Thechnology Division of Mathematics Thessaloniki, 540 06, GREECE