MATEMATIQKI VESNIK 49 (1997), 3–14

UDK 514.763.5

originalni nauqni rad research paper

ON TWO TENSOR FIELDS WHICH ARE ANALOGOUS TO THE CURVATURE AND TORSION TENSOR FIELDS

Kostadin Trenčevski

1. Introducing of two tensor fields

Let M_{n} be a differentiable manifold, and let ξ = (E, π, M_{n}) be a vector bundle of rank k, endowed with a linear connection Γ̃. Apart from the linear connection we suppose that M_{n} is endowed with a linear connection Γ on the tangent bundle. So there exist two covariant derivations ∇ and ˜∇, two curvature tensors

R(X, Y) = [∇_{X}, ∇_{Y}] − ∇_{[X,Y]},
R̃(X, Y) = [ ˜∇_{X}, ˜∇_{Y}] − ˜∇_{[X,Y]}

and one torsion tensor

T (X, Y) = ∇_{Y} X − ∇_{X}Y + [X, Y].

We introduce the following notation

˜∇_{(X,Y)} = R̃(X, Y) + ˜∇_{T (X,Y)}

for X, Y ∈ χ(M_{n}). Now we define a mapping R^{∗}(X, Y, P, Q): Γ̃ξ → Γ̃ξ for given X, Y, P, Q ∈ χ(M_{n}) as follows

R^{∗}(X, Y, P, Q) = ˜∇_{(X,Y)} ^{∘} ˜∇_{(P,Q)} − ˜∇_{(P,Q)} ^{∘} ˜∇_{(X,Y)} − ˜∇_{[T (X,Y),T (P,Q)]}

+ R̃(∇_{T (P,Q)}X, Y) + R̃(X, ∇_{T (P,Q)}Y) − R̃(∇_{T (X,Y)}P, Q) − R̃(P, ∇_{T (X,Y)}Q)

+ R̃(R(P, Q)X, Y) + R̃(X, R(P, Q)Y) − R̃(R(X, Y)P, Q) − R̃(P, R(X, Y)Q). (1.1)

Hence it follows that R^{∗}(X, Y, P, Q): Γ̃ξ → Γ̃ξ is a linear mapping such that R^{∗}(f_{1}X, f_{2}Y, f_{3}P, f_{4}Q) = f_{1}f_{2}f_{3}f_{4}R^{∗}(X, Y, P, Q) for arbitrary functions f_{i} ∈ F(M_{n}) (1 ⩽ i ⩽ 4) and arbitrary vector fields X, Y, P, Q ∈ χ(M_{n}), and thus R^{∗} is a tensor field. Further we will find the components of the tensor field R^{∗} in a local coordinate system.

AMS Subject Classification: 53 B 05

Keywords and phrases: Curvature, torsion, forms

Communicated at the 11. Yugoslav Geometrical Seminar, Divčibare 1996.

Our convenience will be that the Latin indices take their values from the set {1,…, n}, and the Greek indices take their values from the set {1,…, k}. Let X = ∂/∂x^{i}, Y = ∂/∂x^{j}, P = ∂/∂x^{r}, Q = ∂/∂x^{s}. From (1.1) it follows that

(R^{∗}(X, Y, P, Q)s)^{α} = R_{βijrs}^{∗α}s^{β},

for arbitrary section s in the vector bundle, where

R_{βijrs}^{∗α} = R̃_{γij}^{α}R̃_{βrs}^{γ} − R̃_{γrs}^{α}R̃_{βij}^{γ} + R̃_{βpj}^{α}R_{irs}^{p} + R̃_{βip}^{α}R_{jrs}^{p}

− R̃_{βps}^{α}R_{rij}^{p} + R̃_{βrp}^{α}R_{sij}^{p} + R̃_{βrs;p}^{α}T_{ij}^{p} − R̃_{βij;p}^{α}T_{rs}^{p} + R̃_{βpq}^{α}T_{ij}^{p}T_{rs}^{q}. (1.2)

From the definition (1.1) we obtain the following properties

R^{∗}(X, Y, P, Q) = −R^{∗}(Y, X, P, Q),
R^{∗}(X, Y, P, Q) = −R^{∗}(X, Y, Q, P),
R^{∗}(X, Y, P, Q) = −R^{∗}(P, Q, X, Y).

Analogously to (1.1) we can also define the following tensor

R̃^{∗}(X, Y, P, Q) = R̃(X, Y)^{∘} R̃(P, Q) − R̃(P, Q)^{∘} R̃(X, Y)  (1.3)

which will be of special interest in the next sections. In local coordinate system, the components of this tensor are

R̃_{βijrs}^{∗α} = R̃_{γij}^{α}R̃_{βrs}^{γ} − R̃_{γrs}^{α}R̃_{βij}^{γ}.  (1.4)

While the tensor R^{∗} depends on the linear connection of the vector bundle ξ, the tensor R̃^{∗} depends on the connection of the vector bundle ξ only, and it can be obtained from the tensor R^{∗} as a special case by putting T = 0 and R = 0.

Now we will introduce the second tensor field, which is analogous to the torsion tensor field. Let us define a mapping T^{∗}: χ(M_{n}) × χ(M_{n}) × χ(M_{n}) × χ(M_{n}) → χ(M_{n}), by

T^{∗}(X, Y, P, Q) =

= T (∇_{T (P,Q)}X, Y) + T (X, ∇_{T (P,Q)}Y) − T (∇_{T (X,Y)}P, Q) − T (P, ∇_{T (X,Y)}Q) + T (R(P, Q)X, Y) + T (X, R(P, Q)Y) − T (R(X, Y)P, Q) − T (P, R(X, Y)Q). (1.5)

T^{∗} is a linear mapping such that

T^{∗}(f_{1}X, f_{2}Y, f_{3}P, f_{4}Q) = f_{1}f_{2}f_{3}f_{4}T^{∗}(X, Y, P, Q)

for arbitrary functions f_{i} ∈ F(M_{n}) (1 ⩽ i ⩽ 4) and arbitrary vector fields X, Y, P, Q ∈ χ(M_{n}), and hence T^{∗} is a tensor field. This tensor depends on the linear connection Γ only. Similarly to the tensor field R^{∗}, the components of the tensor field T^{∗} in local coordinates are given by

T_{ijrs}^{∗m} = T_{rs;p}^{m}T_{ij}^{p} − T_{ij;p}^{m}T_{rs}^{p} + T_{pq}^{m}T_{ij}^{p}T_{rs}^{q}

+ T_{pj}^{m}R_{irs}^{p} + T_{ip}^{m}R_{jrs}^{p} − T_{ps}^{m}R_{rij}^{p} − T_{rp}^{m}R_{sij}^{p}, (1.6)

and it is obvious that

T^{∗}(X, Y, P, Q) = −T^{∗}(Y, X, P, Q),
T^{∗}(X, Y, P, Q) = −T^{∗}(X, Y, Q, P),
T^{∗}(X, Y, P, Q) = −T^{∗}(P, Q, X, Y).

To the end of this section we will consider the Ricci identities, and we will see that the tensors R^{∗} and T^{∗} naturally appear. Let A_{β_{1}...β_{q}}^{α_{1}...α_{p}} be components of a ξ-tensor field A in the vector bundle ξ. We introduce the following notations for the antisymmetric covariant differentiation

A_{β_{1}...β_{q};(r,s)}^{α_{1}...α_{p}} = A_{β_{1}...β_{q};r;s}^{α_{1}...α_{p}} − A_{β_{1}...β_{q};s;r}^{α_{1}...α_{p}},  (1.7)
A_{β_{1}...β_{q};((ij)(rs))}^{α_{1}...α_{p}} = A_{β_{1}...β_{q};(ij);(rs)}^{α_{1}...α_{p}} − A_{β_{1}...β_{q};(rs);(ij)}^{α_{1}...α_{p}}.  (1.8)

Indeed, ; (ij) is a differentiation and using the tensors R^{∗} and T^{∗}, the Ricci identity is given by

A_{β_{1}...β_{q};((ij)(rs))}^{α_{1}...α_{p}} = − ∑_{u=1}^{p} A_{β_{1}...β_{q}}^{α_{1}...λ...α_{p}}R_{λijrs}^{∗α_{u}}

+ ∑_{u=1}^{q} A_{β_{1}...λ...β_{q}}^{α_{1}...α_{p}}R_{β_{u}ijrs}^{∗λ} − A_{β_{1}...β_{q};u}^{α_{1}...α_{p}}T_{ijrs}^{∗u}. (1.9)

Although it is easy to obtain the formulas (1.2) and (1.6) starting from (1.9), it is not so easy to obtain the formulas (1.1) and (1.5). In the next sections we will consider the geometric interpretation and we will see that these two tensors satisfy analogous formulas as the ordinary curvature and torsion tensors.

2. Geometrical interpretation

In this section a geometrical interpretation of the tensor T^{∗}, antisymmetric differentiation (1.8) and the tensor R^{∗} will be given.

Let X, Y, P, Q ∈ T_{O}(M_{n}) and let τ_{1}^{(1)}(t) (0 ⩽ t ⩽ Δu), τ_{2}^{(1)}(t) (0 ⩽ t ⩽ Δv),τ_{3}^{(1)}(t) (0 ⩽ t ⩽ Δu) and τ_{4}^{(1)}(t) (0 ⩽ t ⩽ Δv) be geodesics which connect the points O and B, B and C, C and D, D and E respectively (fig. 1), such that

(dτ_{1}^{(1)}(t)/dt)_{t=0} = X,  (dτ_{2}^{(1)}(t)/dt)_{t=0} = Y_{B},
(dτ_{3}^{(1)}(t)/dt)_{t=0} = −X_{C},  (dτ_{4}^{(1)}(t)/dt)_{t=0} = −Y_{D},

Fig. 1

where Y_{B} is vector at B obtained by a parallel displacement along the curve τ_{1}^{(1)} using the linear connection Γ, and analogously X_{C} and Y_{D} are defined.

If A ∈ C^{2} is a tensor field in a neighborhood of O and if X = ∂/∂x^{p} and Y = ∂/∂x^{s}, then

A_{β_{1}...β_{q};(rs)}^{α_{1}...α_{p}} = − lim_{ΔU,Δv→0} \frac{1}{ΔuΔv} [φ_{E}^{O}(A_{β_{1}...β_{q}}^{α_{1}...α_{p}}) − (A_{β_{1}...β_{q}}^{α_{1}...α_{p}})_{O}] ,  (2.1)

and it gives the geometrical interpretation of the diferentiation (1.7).

Now we will give similar geometrical interpretation of the differentiation (1.8). We transport the vectors P and Q parallelly to the point E through the points B, C, D, and using the vectors P_{E} and Q_{E} similarly we construct geodesics τ_{1}^{(2)}(t) (0 ⩽ t ⩽ Δu′), τ_{2}^{(2)}(t) (0 ⩽ t ⩽ Δv′), τ_{3}^{(2)}(t) (0 ⩽ t ⩽ Δu′), τ_{4}^{(2)}(t) (0 ⩽ t ⩽ Δv′), and obtain a point F. Similarly, using the parallelly transported vectors −X_{F} and −Y_{F} we can construct geodesics τ_{1}^{(3)}, τ_{2}^{(3)}, τ_{3}^{(3)} and τ_{4}^{(3)} and obtain a point G, and finally, using the vectors −P_{G} and −Q_{G} we find a point H. Further we are going to find the difference (x^{t})_{H} −(x^{t})_{O} of the coordinates of the points H and O at fourth order approximation, neglecting all the terms of order higher than four everywhere.

We notice that

(x^{t})_{E} − (x^{t})_{O} = T_{rs}^{t}X^{r}Y^{s}ΔuΔv + ϑ_{3}(X, Y) + ϑ_{4}(X, Y),  (2.2)
(x^{t})_{F} − (x^{t})_{E} = (T_{rs}^{t})_{E}(P^{r})_{E}(Q^{s})_{E}Δu′Δv′ + ϑ_{3}(P_{E}, Q_{E}) + ϑ_{4}(P_{E}, Q_{E}),(2.3)

where ϑ_{i} (i = 3, 4) are the terms of i-th order of approximation, and (T_{rs}^{t})_{E} are the components of the torsion tensor at the point E. At the second order of approximation it holds

(T_{rs}^{t})_{E} = T_{rs}^{t} + (∂T_{rs}^{t}/∂x^{j})T_{pq}^{j}X^{p}Y^{q}ΔuΔv,  (2.4)

and from (2.1) at the same order of approximation it holds

(P^{r})_{E} = P^{r} − P^{i}(R_{ipq}^{r} + Γ_{ij}^{r}T_{pq}^{j})X^{p}Y^{q}ΔuΔv,  (2.5)
(Q^{s})_{E} = Q^{s} − Q^{i}(R_{ipq}^{s} + Γ_{ij}^{s}T_{pq}^{j})X^{p}Y^{q}ΔuΔv.  (2.6)

By adding the equalities (2.2) and (2.3), and using the equalities (2.4), (2.5) and (2.6), we obtain

(x^{t})_{F} − (x^{t})_{O} = T_{rs}^{t}X^{r}Y^{s}ΔuΔv + T_{rs}^{t}P^{r}Q^{s}Δu′Δv′

+ U_{ijrs}^{t}X^{i}Y^{j}P^{r}Q^{s}ΔuΔvΔu′Δv′ + ϑ_{3}(X, Y) + ϑ_{4}(X, Y) + ϑ_{3}(P, Q) + ϑ_{4}(P, Q),

where

U_{ijrs}^{t} = ∂T_{rs}^{t}/∂x^{p} · T_{ij}^{p} − T_{rp}^{t}(R_{sij}^{p} + Γ_{sq}^{p}T_{ij}^{q}) − T_{ps}^{t}(R_{rij}^{p} + Γ_{rq}^{p}T_{ij}^{q}).

In the same way as the point F was obtained using vectors X, Y, P and Q, another point F′ can be obtained using vectors P, Q, X and Y. Moreover, at the fourth order of approximation it holds

(x^{t})_{H} − (x^{t})_{O} = [(x^{t})_{F} − (x^{t})_{O}] − [(x^{t})_{F′} − (x^{t})_{O}].

Hence and using the following equality

(x^{t})_{F′} − (x^{t})_{O} = T_{rs}^{t}P^{r}Q^{s}Δu′Δv′ + T_{rs}^{t}X^{r}Y^{s}ΔuΔv

+ U_{ijrs}^{t}P^{i}Q^{j}X^{r}Y^{s}ΔuΔvΔu′Δv′ + ϑ_{3}(P, Q) + ϑ_{4}(P, Q) + ϑ_{3}(X, Y) + ϑ_{4}(X, Y),

one can prove that

(x^{t})_{H} − (x^{t})_{O} = T_{ijrs}^{∗t}X^{i}Y^{j}P^{r}Q^{s}ΔuΔvΔu′Δv′,  (2.7)

which gives a geometrical interpretation of the tensor T^{∗}. Especially, if X = ∂/∂x^{i}, Y = ∂/∂x^{j}, P = ∂/∂x^{r} and Q = ∂/∂x^{s}, we obtain

(x^{t})_{H} − (x^{t})_{O} = T_{ijrs}^{∗t}ΔuΔvΔu′Δv′.  (2.8)

Further we will obtain the geometrical interpretation of the differentiation (1.8). In a neighborhood of the point O we choose a ξ-vector field A^{α} of class C^{4}. We parallelly transport the vector A^{α} from the point O to the point E along the previously mentioned geodesic lines, using the connection Γ̃. Then the fourth order of approximation leads to the following transported vector

(A^{α})_{E} = A^{α} − A^{β}(R_{βpq}^{α} + Γ_{βt}^{α}T_{pq}^{t})X^{p}Y^{q}ΔuΔv + ϑ_{3}(X, Y, A) + ϑ_{4}(X, Y, A). (2.9)

Similarly to (2.9) the parallelly transported vector (A^{α})_{E} from E to F is

(A^{α})_{F} = (A^{α})_{E} − (A^{β})_{E}[(R_{βrs}^{α})_{E} + (Γ_{βj}^{α})_{E}(T_{rs}^{j})_{E}](P^{r})_{E}(Q^{s})_{E}Δu′Δv′ + ϑ_{3}(P, Q, A) + ϑ_{4}(P, Q, A),

8 K. Trenčevski

where

(R_{βrs}^{α})_{E} = R_{βrs}^{α} + ∂R_{βrs}^{α}/∂x^{a} · T_{pq}^{a}X^{p}Y^{q}ΔuΔv,
(Γ_{βj}^{α})_{E} = Γ_{βj}^{α} + ∂Γ_{βj}^{α}/∂x^{a} · T_{pq}^{a}X^{p}Y^{q}ΔuΔv,
(T_{rs}^{j})_{E} = T_{rs}^{j} + ∂T_{rs}^{j}/∂x^{a} · T_{pq}^{a}X^{p}Y^{q}ΔuΔv,

(P^{r})_{E}, (Q^{s})_{E} and (A^{α})_{E} are given by (2.5), (2.6) and (2.9). Hence, at the fourth order of approximation we obtain (A^{α})_{F} = A^{α} − A^{β}(R_{βpq}^{α} + Γ_{βj}^{α}T_{pq}^{j})X^{p}Y^{q}ΔuΔv

− A^{β}(R_{βpq}^{α} + Γ_{βj}^{α}T_{pq}^{j})P^{p}Q^{q}Δu′Δv′ + A^{β}V_{βijrs}^{α}X^{i}Y^{j}P^{r}Q^{s}ΔuΔvΔu′Δv′

+ ϑ_{3}(X, Y, A) + ϑ_{4}(X, Y, A) + ϑ_{3}(P, Q, A) + ϑ_{4}(P, Q, A),

where V_{βijrs}^{α} = R_{γrs}^{α}R_{βij}^{γ} + R_{γrs}^{α}Γ_{βp}^{γ}T_{ij}^{p} + R_{βij}^{γ}Γ_{γp}^{α}T_{rs}^{p} + Γ_{βp}^{γ}Γ_{γq}^{α}T_{ij}^{p}T_{rs}^{q} + R_{βps}^{α}R_{rij}^{p}

+ Γ_{βp}^{α}T_{qs}^{p}R_{rij}^{q} + R_{βps}^{α}Γ_{rq}^{p}T_{ij}^{q} + Γ_{βp}^{α}T_{qs}^{p}Γ_{rt}^{q}T_{ij}^{t} + R_{βrp}^{α}R_{sij}^{p} + Γ_{βp}^{α}T_{rq}^{p}R_{sij}^{q} +R_{βrp}^{α}Γ_{sq}^{p}T_{ij}^{q}+Γ_{βp}^{α}T_{rq}^{p}Γ_{st}^{q}T_{ij}^{t}−∂R_{βrs}^{α}/∂x^{p}·T_{ij}^{p}−Γ_{βp}^{α}∂T_{rs}^{p}/∂x^{q}·T_{ij}^{q}−∂Γ_{βp}^{α}/∂x^{q}·T_{rs}^{p}T_{ij}^{q}. Symmetrically, it holds (A^{α})_{F′} = A^{α} − A^{β}(R_{βpq}^{α} + Γ_{βj}^{α}T_{pq}^{j})P^{p}Q^{q}Δu′Δv′

− A^{β}(R_{βpq}^{α} + Γ_{βj}^{α}T_{pq}^{j})X^{p}Y^{q}ΔuΔv + A^{β}V_{βijrs}^{α}P^{i}Q^{j}X^{r}Y^{s}ΔuΔvΔu′Δv′

+ ϑ_{3}(P, Q, A) + ϑ_{4}(P, Q, A) + ϑ_{3}(X, Y, A) + ϑ_{4}(X, Y, A).

Thus, at the fourth order of approximation we obtain (A^{α})_{H} = (A^{α})_{O} +[(A^{α})_{F} −(A^{α})_{F′}] = A^{β}(V_{βijrs}^{α} −V_{βrsij}^{α})X^{i}Y^{j}P^{r}Q^{s}ΔuΔvΔu′Δv′.

According to (2.8), the ξ-vector field A^{α} at the point H has coordinates A^{α} + ∂A^{α}/∂x^{p} · T_{ijrs}^{∗p}X^{i}Y^{j}P^{r}Q^{s}ΔuΔvΔu′Δv′.

Subtracting the parallelly transported vector (A^{α})_{H} from this vector, we obtain the following vector (A_{;p}^{α}T_{ijrs}^{∗p} + A^{β}R_{βijrs}^{∗α})X^{i}Y^{j}P^{r}Q^{s}ΔuΔvΔu′Δv′ =

= −A_{;((ij)(rs))}^{α}X^{i}Y^{j}P^{r}Q^{s}ΔuΔvΔu′Δv′.

Especially, if X = ∂/∂x^{i}, Y = ∂/∂x^{j}, P = ∂/∂x^{r} and Q = ∂/∂x^{s}, we obtain

A_{;((ij)(rs))}^{α} = − lim_{Δu,Δv,Δu′,Δv′→0} \frac{1}{ΔuΔvΔu′Δv′} [(A^{α})_{H} − φ_{H}^{O}(A^{α})],  (2.10)

where φ_{H}^{O} denotes the parallel displacement from O to H with respect to the connection Γ̃, and (A^{α})_{H} denotes the vector of the field A^{α} at the point H. Now (2.10) represents the geometrical interpretation of the differentiation (1.8). Note that (2.10) also holds if the vector field A is substituted by an arbitrary ξ-tensor field.

Further we will give geometrical interpretation of the tensor R̃^{∗}. Let X, Y, P, Q ∈ T_{O}(M_{n}), and let τ′ and τ′′ be two infinitesimal parallelograms with adjacent edges Δu·X and Δv ·Y for τ′, and Δu′ ·P and Δv′ ·Q for τ′′. Let A be an arbitrary ξ-tensor field in a neighborhood of O. We will denote by φ(A) the ξ-tensor at the point O which is obtained by parallel displacement of the ξ-tensor A_{O} along the curve τ′ · τ′′ · (τ′)^{−1} · (τ′′)^{−1} using the connection Γ̃. It is easy to verify that at the fourth order of approximation, it holds

(φ(A))_{β_{1}...β_{q}}^{α_{1}...α_{p}} − A_{β_{1}...β_{q}}^{α_{1}...α_{p}} = [− ∑_{w=1}^{p} A_{β_{1}...β_{q}}^{α_{1}...γ...α_{p}} R̃_{γijrs}^{∗α_{w}}

+ ∑_{w=1}^{q} A_{β_{1}...γ...β_{q}}^{α_{1}...α_{p}} R̃_{β_{w}ijrs}^{∗γ}]X^{i}Y^{j}P^{r}Q^{s}ΔuΔvΔu′Δv′, (2.11)

which gives the required geometrical interpretation.

3. Structure equation and Bianchi identity

Further we will not use the linear connection Γ and the tensors R^{∗} and T^{∗}, and so our convenience will be to use the notations Γ, ∇, R and R^{∗} instead of Γ̃, ˜∇, R̃ and R̃^{∗}.

First we will obtain the structure equation and the Bianchi identity for the curvature tensor R^{∗} in the case of the vector bundle (E, π, M_{n}), and further we will consider the general case when a connection is given in a principal bundle (E, π, E/G).

Like in the non-commutative differential geometry [1], the need of introducing of differential forms of the following kind naturally appears here. Differential rform is a correspondence of an element of Λ^{r}(T′_{x}(M_{n})) to each point x ∈ M_{n}, where T′_{x}(M_{n}) is the vector space of bivectors on T_{x}(M_{n}), i.e.

T′_{x}(M_{n}) = \{ a^{ij}[(\frac{∂}{∂x^{i}} )_{x} ∧ (\frac{∂}{∂x^{j}} )_{x}] : a^{ij} ∈ R \}.

For example, an arbitrary differential 3-form in local coordinates has the following form

w = f_{i_{1}j_{1}i_{2}j_{2}i_{3}j_{3}}(dx^{i_{1}} ∧ dx^{j_{1}}) ∧ (dx^{i_{2}} ∧ dx^{j_{2}}) ∧ (dx^{i_{3}} ∧ dx^{j_{3}}).

For the sake of simplicity our convenience is to write the differential forms as in the following form w = f_{i_{1}j_{1}i_{2}j_{2}i_{3}j_{3}}dx^{i_{1}j_{1}} ∧ dx^{i_{2}j_{2}} ∧ dx^{i_{3}j_{3}}. Note that

dx^{ij} (\frac{∂}{∂x^{i}} ∧ \frac{∂}{∂x^{j}} ) = δ_{p}^{i}δ_{q}^{j} − δ_{p}^{i}δ_{q}^{j}.

The need of consideration of differential forms with values in the vector bundle T_{q}^{p}ξ also appears. Using the tensor R_{βij}^{α} we obtain the following differential 1-form

Ω_{β}^{α} = \frac{1}{2}R_{βij}^{α} dx^{ij}  (3.1)

with values in End ξ = ξ^{∗} ⊗ ξ.

Now we define an operator D^{∗} as follows. If ϕ is a differential r-form with values in T_{q}^{p}ξ, then D^{∗}ϕ is a differential (r + 1)-form with values in the tensor bundle T_{q}^{p}ξ and

(D^{∗}ϕ)_{β_{1}...β_{q}}^{α_{1}...α_{p}} = ∑_{s=1}^{p} Ω_{λ}^{α_{s}} ∧ ϕ_{β_{1}...β_{q}}^{α_{1}...λ...α_{p}} − ∑_{s=1}^{q} Ω_{β_{s}}^{λ} ∧ ϕ_{β_{1}...λ...β_{q}}^{α_{1}...α_{p}}.  (3.2)

The connection between the operator D^{∗} and the differential operator D is given as follows. If

ψ_{β_{1}...β_{q}}^{α_{1}...α_{p}} = A_{β_{1}...β_{q}s_{1}...s_{r}}^{α_{1}...α_{p}} dx^{s_{1}} ∧ ⋯ ∧ dx^{s_{r}}

is an ordinary r-form (i.e. an element of Λ^{r}(T (M_{n}))) with values in T_{q}^{p}ξ, then it is well known that

(Dψ)_{β_{1}...β_{q}}^{α_{1}...α_{p}} = [∂A_{β_{1}...β_{q}s_{1}...s_{r}}^{α_{1}...α_{p}}/∂x^{u} + ∑_{s=1}^{p} Γ_{λu}^{α_{s}}A_{β_{1}...β_{q}s_{1}...s_{r}}^{α_{1}...λ...α_{p}}

− ∑_{s=1}^{q} Γ_{β_{s}u}^{λ}ϕ_{β_{1}...λ...β_{q}s_{1}...s_{r}}^{α_{1}...α_{p}}] dx^{u} ∧ dx^{s_{1}} ∧ ⋯ ∧ dx^{s_{r}}. (3.3)

It can be verified that

(DDψ)_{β_{1}...β_{q}}^{α_{1}...α_{p}} = ∑_{s=1}^{p} Ω_{λ}^{α_{s}} ∧ ψ_{β_{1}...β_{q}}^{α_{1}...λ...α_{p}} − ∑_{s=1}^{q} Ω_{β_{s}}^{λ} ∧ ψ_{β_{1}...λ...β_{q}}^{α_{1}...α_{p}},  (3.4)

where Ω_{β}^{α} is the differential 2-form of curvature. According to (3.2) and (3.4) it seems to be that D^{2} = D^{∗}, but it is not because D^{2} acts on the elements of Λ^{r}(T (M_{n})) but D^{∗} acts on the elements of Λ^{r}(T′(M_{n})).

The definition (3.2) of D^{∗} gives the following properties:

(i) If p = q = 0, then D^{∗}ϕ = 0.

(ii) The operator D^{∗} commutes with the contraction.

(iii) If ϕ_{1} is a differential r-form with values in T_{q}^{p}ξ and ϕ_{2} is a differential s-form with values in T_{v}^{u}ξ, then

D^{∗}(ϕ_{1} ∧ ϕ_{2}) = (D^{∗}ϕ_{1}) ∧ ϕ_{2} + (−1)^{r}ϕ_{1} ∧ (D^{∗}ϕ_{2}).

The operator D^{2} satisfies the properties (i) and (ii) also, and instead of (iii) it satisfies

D^{2}(ϕ_{1} ∧ ϕ_{2}) = (D^{2}ϕ_{1}) ∧ ϕ_{2} + ϕ_{1} ∧ (D^{2}ϕ_{2}).

So D^{∗} is more convenient than D^{2}. Using the tensor R^{∗} we obtain a differential 2-form with values in End ξ = ξ^{∗} ⊗ ξ, as follows

Ω_{β}^{∗α} = \frac{1}{2}R_{βijrs}^{∗α}dx^{ij} ∧ dx^{rs}.  (3.5)

From (3.1), (3.5) and (1.4) it is easy to obtain

Ω_{β}^{∗α} = Ω_{γ}^{α} ∧ Ω_{β}^{γ}  (3.6)

and

Ω_{β}^{∗α} = D^{∗}(Ω_{β}^{α}).  (3.7)

The equation (3.7) is the structural equation for the form of curvature Ω^{∗}. Note that if Ω_{β}^{α} was considered as a differential 2- form (an element of Λ^{2}(T (M_{n}))) we would obtain DD(Ω_{β}^{α}) = 0. That shows the need of consideration of new class of differential forms as elements of Λ^{r}(T′(M_{n})).

From (3.2) and (3.6) it follows that

D^{∗}Ω_{b}^{∗α} = 0  (3.8)

which is the Bianchi identity for the curvature form Ω^{∗}.

If ϕ is a differential r-form, then (3.2) and (3.6) imply the following identity

(D^{∗}D^{∗}ψ)_{β_{1}...β_{q}}^{α_{1}...α_{p}} = ∑_{s=1}^{p} Ω_{λ}^{∗α_{s}} ∧ ψ_{β_{1}...β_{q}}^{α_{1}...λ...α_{p}} − ∑_{s=1}^{q} Ω_{β_{s}}^{∗λ} ∧ ψ_{β_{1}...λ...β_{q}}^{α_{1}...α_{p}},  (3.9)

which has the same form as the operator D · D. So we can introduce a new class of differential forms which correspond an element of Λ^{r}(T′′_{x}(M_{n})) to each point x ∈ M_{n}, where T′′_{x} = T′_{x}(T′_{x}(M_{n})). Thus

Ω_{β}^{∗α} = \frac{1}{2}R_{βijrs}^{∗α}dx^{ijrs}  (3.10)

is a differential 1-form with values in End ξ and for arbitrary r-form ϕ with values in T_{q}^{p}ξ, we define the following differentiation

(D^{∗∗}ϕ)_{β_{1}...β_{q}}^{α_{1}...α_{p}} = \frac{1}{2}[∑_{s=1}^{p} Ω_{λ}^{∗α_{s}} ∧ ϕ_{β_{1}...β_{q}}^{α_{1}...λ...α_{p}} − ∑_{s=1}^{q} Ω_{β_{s}}^{∗λ} ∧ ϕ_{β_{1}...λ...β_{q}}^{α_{1}...α_{p}}].  (3.11)

This procedure can be further continued, and the corresponding structural equations and the Bianchi identities can be obtained.

Now we will generalize the previous considerations to the case of a principal Gbundle (E, π, E/G). Let ω be the form of connection and Ω be the form of curvature. Then it is known that the tensorial forms of type Ad G can be identified by ξ[Ad]- valued forms on M_{n} = E/G, where ξ[Ad] is the associated vector bundle ([3]). Thus we will use the tensorial forms of type Ad G on E, instead of ξ[Ad]-valued forms on M_{n}.

Let ψ be an ordinary tensorial r-form of type Ad G, in sense that at each point u ∈ E we have an element of Λ^{r}(T_{u}(E)). Then it is well known that

Dψ = dψ + [ω, ψ].  (3.12)

From the definition of the operation [ ], the following identities can be verified

d[ψ_{1}, ψ_{2}] = [dψ_{1}, ψ_{2}] − [ψ_{1}, dψ_{2}]  (3.13)

and

[ψ_{1}, [ψ_{1}, ψ_{2}]] = \frac{1}{2}[[ψ_{1}, ψ_{1}], ψ_{2}],  (3.14)

where ψ_{1} is a differential 1-form and ψ_{2} is a differential r-form. Using the formulas (3.12), (3.13) and (3.14), if ϕ is an arbitrary r-form of type Ad G, we obtain

(D · D)ψ = D(dψ + [ω, ψ]) = d[ω, ψ] + [ω, dψ + [ω, ψ]]
= [dω, ψ] − [ω, dψ] + [ω, [ω, ψ]] + [ω, dψ]
= [dω, ψ] + \frac{1}{2}[[ω, ω], ψ] = [Ω, ψ].  (3.15)

Further we are going to define tensorial forms of type Ad G in sense that for each point u ∈ E we have an element of Λ^{r}(T′_{u}(E)). So we first define that the bivector S is horizontal, if it can be represented in the following form

S = ∑_{i=1}^{n} ∑_{j=1}^{n} a_{ij}X_{i} ∧ X_{j},

where X_{1}, …, X_{n} are horizontal vectors. Besides, it is naturally to define

(R_{α})_{∗}(Y_{1} ∧ Y_{2}) = (R_{α})_{∗}Y_{1} ∧ (R_{α})_{∗}Y_{2},

where Y_{1}, Y_{2} ∈ T_{u}(E). Now we define a new class of tensorial r-forms of type Ad G as g-valued differential r-forms, which correspond to each point u ∈ E an element of Λ^{r}(T′_{u}(E)) such that:

(i) R_{a}^{∗}ϕ = (Ad a^{−1})ϕ for each a ∈ G, and

(ii) ϕ(S_{1},…, S_{r}) = 0, if at least one of the bivectors S_{1}, …, S_{r} is not horizontal.

Now the tensorial form of the curvature can be interpreted as a tensorial 1form Ω′ of type Ad G in the sense of the previous definition, if we suppose that Ω′(X ∧ Y) = Ω(X, Y). Thus, if ϕ is a tensorial r-form of type Ad G in the sense of the previous definition, then the formula (3.15) leads us to introduce an operator D^{∗} by

D^{∗}ϕ = [Ω′, ϕ]  (3.16)

and now D^{∗}ϕ is a tensorial (r + 1)-form of type Ad G. From the definition of the operation [ ] one can verify that

D^{∗}(ϕ_{1} ∧ ϕ_{2}) = (D^{∗}ϕ_{1}) ∧ ϕ_{2} + (−1)^{r}ϕ_{1} ∧ (D^{∗}ϕ_{2}),

where ϕ_{1} is a tensorial r-form and ϕ_{2} is a tensorial s-form of type Ad G. Further we define tensorial 2-form of curvature Ω^{∗} by Ω^{∗} = [Ω′, Ω′], i.e.

Ω^{∗} = D^{∗}(Ω′).  (3.17)

The formula (3.17) represents the structural equation. It is easy to verify that

D^{∗}(Ω^{∗}) = 0,  (3.18)

which is the Bianchi identity.

The equality (3.14) implies

(D^{∗} · D^{∗})ϕ = \frac{1}{2}[Ω^{∗}, ϕ].  (3.19)

This formula leads us to introduce a new operator, if we consider Ω^{∗} as a tensorial 1form of type Ad G. This procedure can be continued further, and the corresponding structural equations and the Bianchi identities can be obtained.

4. Holonomy group

Let Φ_{0}(u) be the restricted holonomy group for the considered principal bundle (E, π, E/G) at point u ∈ E, and we will denote by f(u) the corresponding Lie algebra. We suppose that E/G is a connected manifold. We will prove the following proposition.

Proposition. Suppose that there exists a point v ∈ E such that f(v) is generated by the elements of the form Ω_{v}(A, B) where A, B ∈ H_{v} are arbitrary horizontal vectors at the point v. If Ω^{∗} iz zero form on E, then Φ_{0}(u) is a commutative group for each u ∈ E.

Conversely, if Φ_{0}(u) is a commutative group, then Ω^{∗} iz a zero form on E.

Proof. Let f(v) is generated by the elements Ω_{v}(A, B), A, B ∈ H_{v} and let Ω^{∗} = 0. Hence

[Ω_{v}(A, B), Ω_{v}(C, D)] = [Ω′_{v}(A ∧ B), Ω′_{v}(C ∧ D)] = \frac{1}{2}[Ω′_{v}, Ω′_{v}](A ∧ B, C ∧ D)

= \frac{1}{2}Ω_{v}^{∗}(A ∧ B, C ∧ D) = 0

for each A, B, C, D ∈ H_{v}, and so [X, Y] = 0 for each X, Y ∈ f(v). Since Φ_{0}(v) is a connected Lie group whose Lie algebra is f(v), it follows that Φ_{0}(v) is a commutative group, and hence Φ_{0}(u) is commutative for each u ∈ E.

Let Φ_{0}(ω) is a commutative group, and let u ∈ E be an arbitrary point. Then Φ_{0}(u) is also a commutative group. So, [X, Y] = 0 for each X, Y ∈ f(u) and hence Ω_{u}^{∗} iz a zero form.

Moreover, if there exists v ∈ E which can be connected with u by a horizontal path and such that f(u) is generated by the elements of the form Ω_{v}(A, B) where A, B ∈ H_{v}, then one can prove that the derived group Φ′_{0}(u) is a Lie group and its Lie algebra is generated by the elements of the form Ω_{v}^{∗}(A ∧ B, C ∧ D) where A, B, C, D ∈ H_{v}.

REFERENCES

[1] A. Connes, Non-Commutative Differential Geometry, Publ. Math. IHES 62 (1985), 41–114.

[2] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, vol. I, Interscience publishers, New York, London 1963.

[3] M. Postnikov, Differential Geometry, Lectures on geometry, semester IV, Nauka, Moskva 1988 (in russian).

14 K. Trenčevski (received 25.10.1996.) Institute of Mathematics, P.O.Box 162, 91000 Skopje, Macedonia e-mail: kostatre@iunona.pmf.ukim.edu.mk