Invariant Operators on Real and Complex Manifolds

Iulia Elena Hirică

Abstract

The development of decomposition theory of the curvature tensors under the action of some groups was initiated in [31], [30]. Since these results and their ideas are very useful in the studies of some problems in geometry and topology of manifolds, many mathematicians have worked using this algebraic treatment of curvature tensors.

Interesting applications of the theory of decompositions are used in the study of submanifolds in conformal differential geometry, classification of almost complex manifolds [6], almost product manifolds [27]. In the Kähler geometry, decompositions of the K-curvature tensors were given in [19], [26], [32], for quaternionic Kähler manifold [35] and the case of contact geometry is studied in [18], [21]. In [20], [23] is investigated the trace decomposition problem. All these splittings are in principle consequences of the general theorems on group representations [44].

This paper develops the original ideas of C.Udrişte regarding the decomposition problem of geometrical objects, introducing some (r, r)-tensor algebras Inv(r), CInv(r), HInv(r), CHInv(r) of operators invariant under certain groups ([13], [15], [36], [37]). These invariant tensor algebras, having the elements interpreted like endomorphisms on the space T_{r−1}^{1}(M), enable us to get a trace decomposition of this space, the results of [20] being special cases of our theory.

The focus is on the infinite subset of projections which do provide good insight in some problems of differentiable manifolds. Let us mention that the Weyl projective curvature tensor, Weyl conformal curvature tensor, H-projective curvature tensor, Bochner curvature tensor and the Thomas projective connection, Thomas conformal connection, H-projective connection and complex conformal connection are produced by this type of operators. Also, using some invariant operators which are projections one gets the splitting of the space of tensors of type (1, 3) into three components invariant under some special groups into infinitely many ways. We should remark that in particular cases, one finds the Strichartz decomposition , Singer-Thorpe-Nomizu decomposition, respectively Sitaramayya-Mori decomposition of the space of curvature tensors.

The extension of the invariant operators to T_{r−1}^{1}(M) and to the F(M)- module A_{r−1}^{1}(M), a new space which is required by our theory, leads us to the problem of decompositions of tensors and connections. Finally, one gets invariants for some transformations of geometrical object fields, extending the Thomas-Weyl theory.

Balkan Journal of Geometry and Its Applications, Vol.4, No.2, 1999, pp. 69-90 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

Along the line developed in [43], [45], we are studied properties of pairs of connections (Γ, Π), where Γ is an affine connection and Π is the Thomas projective connection, respectively Thomas conformal connection or Γ is the Levi-Civita connection and Π is the H-projective related J-connection, respectively complex conformal connection, we investigate some pairs of geometrical object fields associated by certain invariant operators.

The study of geodesically related Riemannian manifolds [43], [41] was extended to semi-symmetric spaces geodesically related in [5] and to subgeodesic correspondence in [28].

We continue the research in this direction studying in the last section some geodesic and subgeodesic mappings [7], [8], [11], [14]. Certain properties of semisymmetric connection on Weyl generalized manifolds are also presented [9], [12].

Mathematics Subject Classification: 53A55, 53B10, 53B20, 53B35 Key words: invariant operators, decomposition problem, pairs of geometrical object fields.

1 Pairs of geometrical object fields via absolute invariant operators

The complete decomposition in Riemannian projective geometry was studied in [1], where is solved the problem under the action of SO(n). The splitting of the space of curvature tensors under the action of GL(V) was obtained in [33]. It is possible to processe and develop the decomposition theory introducing a special algebra of invariant operators, built with the Kronecker tensor.

1.1 Absolut invariant tensors algebra

Let V be a real n-dimensional vector space, where n > 2. Let T_{r}^{r}(V) be the vector space of all tensors of type (r, r) and δ_{j}^{i} be the symbol of Kronecker. In T_{r}^{r}(V)

we consider the vector subspace Inv(r) = {∑_{σ∈S_{r}} x_{σ}δ_{j_{1}}^{i_{σ(1)}}...δ_{j_{r}}^{i_{σ(r)}}|x_{σ} ∈ R, σ ∈ S_{r}} , where S_{r} is the group of permutations. Any element P ∈ Inv(r), which is an absolute invariant tensor (i.e., ∀A ∈ GL(V), A ∘ P = P), is interpreted like an endomorphism on T_{r−1}^{1}(V), producing a trace decomposition of this space. So, Inv(r) becomes an algebra of absolute invariant operators, the product P Q of two elements P, Q being given by the rule P_{j_{1}}^{i_{1}...i_{r−1}} _{j_{2}...j_{r}}^{i_{r}}Q_{i_{r}}^{j_{2}...j_{r}} _{i_{r+1}...i_{2r−1}}^{j_{r+1}}.

It is interesting to study the pair (T, P T), ∀P ∈ Inv(r), T ∈ T_{r−1}^{1}(V), where (P T)(ω, X_{1}, ..., X_{r−1}) = ∑_{σ∈S_{r−1}}[x_{σ}T (ω, X_{σ(1)}, ..., X_{σ(r−1)})+

+ ∑_{k=1}^{r−1} x_{σ}^{k}(C_{k}^{1}T)(X_{σ(1)}, ..., X̂_{σ(k)}, ..., X_{σ(r−1)})ω(X_{σ(k)})], ∀ω ∈ V′, ∀X_{k}, X_{σ(k)} ∈ V, σ ∈ S_{r}, C_{k}^{1} being the contraction map, ∀k ∈ {1, ..., r−1}.

Important geometrical meanings have the cases r = 3, 4, which appear in the study of connections, torsion tensors, curvature tensors, Weyl projective curvature tensors etc.

The operator P = {P_{a}^{bc} _{st}^{r}} ∈ Inv(3) having the simplified expression P = x_{1}I_{1} + ... + x_{6}I_{6}, where {I_{1}, ..., I_{6}} is the basis, acts like un endomorphism on T_{2}^{1}(V). Endowing Inv(3) with a structure of Lie over R, it is possible to clasify its subalgebras Theorem 1.1.1 D_{1} = Span{I_{1}, I_{2}, I_{6}} and D_{2} = Span{I_{1}, I_{2}, I_{5}} are 3-dimensi-onal Lie subalgebras and B_{1} = Span{I_{1}, I_{2}}, B_{2} = Span{I_{1}, I_{3}}, B_{3} = Span{I_{1}, I_{5}} are 2-dimensional Lie subalgebras of the Lie algebra Inv(3).

Moreover, D_{1} and D_{2} are solvable subalgebras, having the abelian ideals R_{1} = {P = αI_{2} − nαI_{6} | α ∈ R}, respectively R_{2} = {P = αI_{2} − nαI_{5} | α ∈ R}.

Any operator P = {P_{a}^{bcd} _{stp}^{r}} ∈ Inv(4) has the simplified expression P = ∑_{i=1}^{24} y_{i}I_{i},

where {I_{1}, ..., I_{24}} is the basis and is interpreted like an endomorphism on T_{3}^{1}(V). The multiplication table PQ, P_{a}^{bcd} _{stp}^{r}Q_{r}^{stp} _{jkl}^{i}, of two elements of Inv(4) was determined using a Borland C++ Programme.

The image P_{a}^{bc} _{st}^{r}T_{bc}^{a}, where P = {P_{a}^{bc} _{st}^{r}} ∈ Inv(3) and T = {T_{bc}^{a}} ∈ T_{2}^{1}(V), represents a traceless decomposition of T. Let P = {P_{i}^{jkl} _{stp}^{r}} ∈ Inv(4) and T = {T_{jkl}^{i}} ∈ T_{3}^{1}(V). The image P_{i}^{jkl} _{stp}^{r}T_{jkl}^{i} is a traceless decomposition of T. The results obtained by D. Krupka, J. Mikesh concerning the traceless decomposition of tensors correspond to some particular cases of our theory. Theorem 1.1.2 There are infinitely many endomorphisms in Inv(3), respectively Inv(4), having traceless images.

1.2 Generalization of Strichartz decomposition

Some absolut invariant operators which are projections do provide insight in some problems of differential geometry. Let us mention that projections from Inv(4) give the Weyl projective curvature tensor and affine transformations of elements from Inv(3) produce the Thomas projective connection. Moreover, properties of the projections enable us to get splittings of the space of tensors of type (1, 3). So, for geometrical reasons, we study the subset of projections of Inv(r). Theorem 1.2.1 There are infinitely many projective projections P ∈ Inv(3), respectively P ∈ Inv(4).

These infinite families of solutions and their geometric representations can be determined using ”MathCad Plus”. We should remark that, for r = 3, in particular if x_{1} = x_{3} = 0, introducing the parameters x_{5} = λ, x_{6} = β, then x_{2} and x_{4}, solutions of the equation z^{2}+ \frac{1}{2}(λ+μ−1)+λμ = 0, belong to one sheet hyperboloids, independent of the dimension of the vector space V.

R.S. Strichartz [33] found a decomposition of K(V), the space of tensors of type (1, 3), verifying R_{stl}^{r} + R_{slt}^{r} = 0 and the first Bianchi identity, using properties of the representation theory.

The projective projections from Span{I_{1}, I_{7}, ..., I_{12}} ⊂ Inv(4) produce infinitely many splittings of the space T_{3}^{1}(V) into three subspaces, invariant under to the group GL(V). Theorem 1.2.2 There are infinitely many nonvanishing projective projections P, Q, R ∈ Span{I_{1}, I_{7}, ..., I_{12}}, such that P + Q + R = I_{1}, PQ = QP = RQ = QR = 0 and T_{3}^{1}(V) = ImP ⊕ ImQ ⊕ ImR holds, where

P = I_{1} + ∑_{i=7}^{12} y_{i}I_{i},

Q = \frac{y_{8} − y_{7}}{2} (I_{7} − I_{9}) + \frac{y_{10} − y_{8}}{2} (I_{8} − I_{10}) + \frac{y_{12} − y_{11}}{2} (I_{12} − I_{11}),

R = −\frac{y_{8} + y_{7}}{2} (I_{7} + I_{9}) − \frac{y_{10} + y_{8}}{2} (I_{8} + I_{10}) − \frac{y_{12} + y_{11}}{2} (I_{12} + I_{11}),

where 1 + ny_{7} + y_{8} + y_{12} = 0, ny_{9} + y_{10} + y_{11} = 0. Proposition 1.2.1 Any tensor R ∈ T_{3}^{1}(V) splitts into infinitely many ways R = R′ +R”+R′′′, such that R′_{ij} = 0, R”_{ij} is a symmetric tensor, R′′′_{ij} is a skew symmetric tensor. Proposition 1.2.2 The endomorphisms P ∈ Span{I_{1}, I_{7}, ..., I_{12}} of the algebra Inv(4) applying K(V) into the same subspace are

P = y_{1}I_{1} + y_{7}I_{7} − y_{7}I_{8} − y_{10}I_{9} + y_{10}I_{10} − (y_{7} + y_{10})I_{11} + (y_{7} + y_{10})I_{12}.

Into this set of infinite family of endomorphisms from Span{I_{1}, I_{7}, ..., I_{12}}, its finite subset of projections has particular geometrical meanings. Remark 1.2.1 In the particular case

P = I_{1} − \frac{n}{n^{2} − 1}I_{7} + \frac{n}{n^{2} − 1}I_{8} − \frac{1}{n^{2} − 1}I_{9} + \frac{1}{n^{2} − 1}I_{10} + \frac{1}{n + 1}I_{11} − \frac{1}{n + 1}I_{12},

Q = \frac{1}{2(n + 1)}(I_{7} − I_{8} − I_{9} + I_{10} − 2I_{11} + 2I_{12}), R = \frac{1}{2(n − 1)}(I_{7} − I_{8} + I_{9} − I_{10})

we find again the Strichartz decomposition [33] K(V) = ImP ⊕ ImQ ⊕ ImR of the curvature tensors into three irreducible components under the action of the group GL(V). This decomposition is not irreducible under O(n) or SO(n).

1.3 δ-decompositions of geometrical object fields

Let M be a differentiable n–dimensional manifold, T_{r}^{r}(M) the bundle of (r, r)–tensor fields of M. Then

Inv(r) = {P_{j_{1}...j_{r}}^{i_{1}...i_{r}} = ∑_{σ∈S_{r}} f_{σ}δ_{j_{1}}^{i_{σ(1)}}...δ_{j_{r}}^{i_{σ(r)}} | f_{σ} ∈ F(M), σ ∈ S_{r}} is the F(M)-module of absolut invariant tensor fields.

We define the F(M)-module denoted A_{r−1}^{1}(M), generated by the union of parallel afine spaces of geometrical object fields of type (1, r − 1) ([24]), whose difference or skew symmetric part with respect to a pair of indices is a tensor field of type (1, r−1). Obviously, for r = 3, the affine space C, of affine connections on M, and the space T_{2}^{1}(M) are examples of such parallel spaces. Each element P of Inv(r) acts like an endomorphism on T_{r−1}^{1}(M) and induces an affine transformation on A_{r−1}^{1}(M), producing trace decomposition of this space.

Along the line developed by T.Y. Thomas, G. Vrănceanu which studied pairs of connections (Γ, Π), where Γ is an affine connection and Π is the projective Thomas connection, we generalize the theory studying the graph (Γ, P Γ) of affine transformations P acting on the space A_{2}^{1}. Remark 1.3.1 If the projective projections act on the affine symmetric connections, then in certain cases, for particular values of the coeficients x_{1}, ..., x_{6}, we get the pair

of connections (Γ, Π), where Π_{st}^{r} = Γ_{st}^{r}− \frac{1}{n + 1}(δ_{s}^{r}Γ_{t}+δ_{t}^{r}Γ_{s}), Γ_{t} = Γ_{at}^{a} is the projective

Thomas connection. Theorem 1.3.1 There are infinitely many nonvanishing triplets P, Q, R, affine transformations on A_{3}^{1}(M), of projective projections from Span_{F(M)}{I_{1}, I_{7}, …, I_{12}}, such that P + Q + R = I_{1}, PQ = QP = RQ = QR = 0. Moreover, A_{3}^{1}(M) = ImP ⊕ ImQ ⊕ ImR.

The relation between the geometrical object fields from A_{2}^{1}(M), respectively A_{3}^{1}(M) is determined by the next closed diagrame which reflects an invariance of gauge type.

Theorem 1.3.2 Let P = x_{1}I_{1} + x_{2}I_{2} + x_{6}I_{6}, P = y_{1}I_{1} + y_{7}I_{7} + y_{8}I_{8} + y_{21}I_{21} + y_{22}I_{22} + y_{24}I_{24}, T = z_{1}I_{1} + z_{7}I_{7} + z_{8}I_{8} + z_{21}I_{21} + z_{22}I_{22} + z_{24}I_{24}, R = I_{1} − I_{2} be affine transformations on A_{2}^{1}(M), respectively A_{3}^{1}(M). There are infinitely many projective projections P, T , P such that next diagrame is closed

(∗)  \begin{aligned}Γ & \underrightarrow{P} & Π \\ ↓ & & ↓ \\ A & & Ā \\ R ↓ & & ↓ R \\ R & & π \\ T ↘ & & ↙ P , \\ & W \end{aligned}

where (Γ, Π) is a pair from A_{2}^{1}(M), Γ being an affine symmetric connection and Π = P Γ the exotic Thomas connection,

(A, Ā) is a pair from A_{3}^{1}(M), associated to (Γ, Π) by the rule A_{stl}^{r} = \frac{∂Γ_{st}^{r}}{∂x^{l}} + Γ_{mt}^{r}Γ_{sl}^{m},

(R, π) is a pair from A_{3}^{1}(M), R being the curvature tensor field associated to Γ and π the exotic curvature tensor field associated to π and W is the exotic projective Weyl curvature tensor field, W = (P + Q) π = (T + S) R. Corollary 1.3.1 In the hypothesis of the previous theorem, if moreover the Ricci tensor associated to the affine connection Γ is symmetric, then the diagrame (∗) is closed for the next cases

1) P = I_{1}, T = P ∈ {0, I_{1}, y_{7}I_{7} + (1 − ny_{7})I_{8}, I_{1} − y_{7}I_{7} + (ny_{7} − 1)I_{8}};

2) P = I_{1} − \frac{1}{n}I_{2}, T = P ∈ \{0, I_{1}, y_{7}I_{7} + (1 − ny_{7})I_{8}, I_{1} − y_{7}I_{7} + (ny_{7} − 1)I_{8}\};

3) P = I_{1} + x_{2}I_{2} − (1 + nx_{2})I_{6}, x_{2} ∈ F(M) \ \{−\frac{1}{n}\},

T = P ∈ \{0, I_{1} − \frac{1}{n − 1}I_{7}+ \frac{1}{n − 1}I_{8}\};

4) P = 0, T = 0, P ∈ {0, I_{1}, y_{7}I_{7} + (1 − ny_{7})I_{8}, I_{1} − y_{7}I_{7} + (ny_{7} − 1)I_{8}};

5) P = \frac{1}{n}I_{2}, T = 0, P ∈ \{0, I_{1}, y_{7}I_{7} + (1 − ny_{7})I_{8}, I_{1} − y_{7}I_{7} + (ny_{7} − 1)I_{8}\};

6) P = x_{2}I_{2} + (1 − nx_{2})I_{6}, x_{2} ∈ F(M) \ \{\frac{1}{n}\}, T = 0, P ∈ \{0, I_{1} − \frac{1}{n − 1}I_{7}+

+\frac{1}{n − 1}I_{8}\} , where y_{7} ∈ F(M).

Taking into account the case 3), we deduce that the proiective Weyl curvature tensor field W is invariant to infinitely many transformations of exotic connections

Γ\underrightarrow{P}Π, P = I_{1} + x_{2}I_{2} − (1 + nx_{2})I_{6}, x_{2} ∈ F(M) \ \{−\frac{1}{n}\} .

The study of the closed diagrame and the geometrical interpretation of the results, when operators P, P, T are generated by all elements of the bases of Inv(3) and Inv(4), remains an open problem.

2 Pairs of geometrical object fields via conformal invariant operators

Our algebra of conformal invariant operators CInv(r) illustrates a general approach of the decomposition problem, in Riemannian geometry, in particular cases corresponding to the Singer-Thorpe-Nomizu splitting of the space of the curvature tensors and also extends the Thomas-Weyl theory.

2.1 Algebra CInv(r) of conformal invariant operators

Let V be a real n-dimensional vector space, n > 2, endowed with the inner product g = (g_{ij}), where g^{−1} = (g^{ij}) and δ = (δ_{j}^{i}) the Kronecker symbol. In the vector space T_{r}^{r}(V) of tensors of type (r, r) we define the vector subspace

CInv(r) = {P, P_{j_{1}...j_{r}}^{i_{1}...i_{r}} = ∑_{σ∈S_{r}} x_{σ}δ_{j_{1}}^{i_{σ(1)}} … δ_{j_{r}}^{i_{σ(r)}} +

∑_{σ(1)<σ(2) , τ(1)<τ(2)} x_{στ} g_{j_{τ(1)}j_{τ(2)}}g^{i_{σ(1)}i_{σ(2)}}δ_{j_{τ(3)}}^{i_{σ(3)}} … δ_{j_{τ(r)}}^{i_{σ(r)}} | x_{σ}, x_{στ} ∈R, σ, τ ∈S_{r}},

where S_{r} is the permutation group. Any element P ∈ CInv(r) is a conformal invariant tensor (i.e. ∀A ∈ CO(V), A ∘ P = P). We endow CInv(r) with a structure of real Lie algebra, as in the subsection 1.1, containing the subalgebra Inv(r).

The algebra CInv(3) arroses during the study of connections, torsion tensors etc. The operators P = {P_{a}^{bc} _{st}^{r}} ∈ Inv(3) are endomorphisms on the space T_{2}^{1}(V), having the simplified expression P = ∑_{i=1}^{6}x_{i}I_{i} + ∑_{j=1}^{9}x_{j+6}G_{j}, where {I_{1}, ..., I_{6}, G_{1}, ..., G_{9}} is the basis.

For r = 4, in the algebra CInv(4) containing the operators P = {P_{a}^{bcd} _{stl}^{r}}, interpreted like endomorphisms on the space T_{3}^{1}(V), having the simplified expression P = ∑_{i=1}^{24}y_{i}I_{i} + ∑_{j=1}^{72} z_{j}G_{j}, {I_{1}, …, I_{24}, G_{1}, …, G_{72}} being the basis, we study for geometrical reasons the subalgebra

D = {P = y_{1}I_{1} + y_{7}I_{7} + y_{8}I_{8} + z_{1}G_{1} + z_{2}G_{2} + z_{3}G_{3} + z_{4}G_{4}}. For particular values of the coeficients, an operator P ∈ D produces the conformal Weyl curvature tensor field. Theorem 2.1.1 Endowing D with a structure of Lie algebra, all its subalgebras are Lie algebras. Remark 2.1.1 There are infinitely many endomorphisms in CInv(3) and CInv(4), having traceless images.

2.2 Generalization of Singer-Thorpe-Nomizu decomposition

For geometrical reasons it is worthwhile to investigate the infinite set of conformal projections on tensors of type (1, r − 1). Indeed, the conformal Weyl curvature tensor field is produced by a projection of the subalgebra D ⊂ CInv(4) and the conformal Thomas connection is built by one affine transformation of a projection from CInv(3).

In [31] I.M. Singer and J.A. Thorpe studied the descomposition of the space R(V) of tensors R of type (1, 3), verifying R_{stl}^{r} + R_{slt}^{r} = 0 and R_{pstl} + R_{sptl} = 0. This is also investigated by K. Nomizu [30], for generalized tensor fields.

Moreover, the space of tensors of type (1, 3) splitts in infinitely many ways into three subspaces, images of some projections from CInv(4). So, using the method of conformal projections, we extend Singer-Thorpe-Nomizu decomposition of curvature tensors. Theorem 2.2.1 There are infinitely many nonvanishing conformal projections P, Q, R ∈ D such that P + Q + R = I_{1}, PQ = QP = RQ = QR = 0. Moreover, the splitting T_{3}^{1}(V) = ImP ⊕ ImQ ⊕ ImR into three invariant subspaces with respect to the group CO(V) holds, where

P = I_{1} + y_{7}I_{7} + y_{8}I_{8} + ∑_{i=1}^{4}z_{i}G_{i}, Q = −\frac{y_{7} + z_{2} + nz_{3}}{n} G_{3} − \frac{y_{8} + z_{1} + nz_{4}}{n} G_{4},

R = −y_{7}I_{7} − y_{8}I_{8} − z_{1}G_{1} − z_{2}G_{2} + \frac{y_{7} + z_{2}}{n} G_{3} + \frac{y_{8} + z_{1}}{n} G_{4},

ny_{7} + y_{8} + z_{1} + 1 = 0, z_{2} + nz_{3} + z_{4} = 0. Proposition 2.2.1. Any tensor R of type (1, 3) splitts into infinitely many possibilities

R = R′ + R” + R′′′ such that R′_{ij} = 0, R”_{ij} = R_{ij} − \frac{1}{n}Kg_{ij}, R′′′_{ij} = \frac{1}{n}Kg_{ij}, where

K = R_{ij}g^{ij} . Remark 2.2.1. In the particular case

P = I_{1}−\frac{1}{n − 2}I_{7}+\frac{1}{n − 2}I_{8}+\frac{1}{n − 2}G_{1}−\frac{1}{n − 2}G_{2}+\frac{1}{(n − 1)(n − 2)}G_{3}−\frac{1}{(n − 1)(n − 2)}G_{4},

R = \frac{1}{n(n − 1)}G_{3} − \frac{1}{n(n − 1)}G_{4},

Q = \frac{1}{n − 2}I_{7} − \frac{1}{n − 2}I_{8} − \frac{1}{n − 2}G_{1} + \frac{1}{n − 2}G_{2} − \frac{2}{n(n − 2)}G_{3} + \frac{2}{n(n − 2)}G_{4},

T_{3}^{1}(V) = ImR ⊕ ImP ⊕ ImQ implies the irreducible orthogonal Singer - Thorpe - Nomizu decomposition under the group O(n) of curvature type tensors

L(V) = L_{1}(V) ⊕ L_{W} (V) ⊕ L_{2}(V), where

L_{1}(V) = {L ∈ L(V) with constant sectional curvature },

L_{1}^{⊥}(V) = {L ∈ L(V) with vanishing scalar curvature },

L_{W} (V) = {L ∈ L(V) with vanishing Ricci tensor },

L_{2}(V) = the orthogonal complement of L_{W} (V) in L_{1}^{⊥}(V),

L(V) = {L ∈ R(V) verifying the first Bianchi identity }.

Our general aprouch is motivated by this particular case.

2.3 δ - g - decompositions of geometrical object fields

Let (M, g) be a Riemannian n - dimensional manifold, T_{r−1}^{1}(M) the F(M)-module of tensor fields of type (1, r − 1) and A_{r−1}^{1}(M) the affine F(M)-module over T_{r−1}^{1}(M), built in the subsection 1.3.

The conformal operators P are extended to T_{r−1}^{1}(M), imposing the condition x_{σ}, x_{στ} ∈ F(M), σ, τ ∈ S_{r} and generate a F(M)-module of invariant endomorphisms under conformal transformations, denoted CInv(r). We denote also with P the induced transformation on the affine F(M)-modulule A_{r−1}^{1}(M), which produces a conformal decomposition of this space.

Let Γ^{∘} be the Levi-Civita connection associated to g and P a conformal projection. We find the pair of geometrical object fields (Γ^{∘}, P Γ^{∘}) of A_{2}^{1}(M), where P Γ^{∘} is called the exotic conformal connection.

For x_{1} = 1, x_{2} = x_{6} = −x_{7} = −\frac{1}{n}, we obtain the pair of connections (Γ^{∘}, P Γ^{∘}),

where P Γ^{∘} is the conformal Thomas connection. This particular case motivates the general method of conformal projections.

Taking into account the study made in the vectorial case, one gets Theorem 2.3.1 There are infinitely many nonvanishing triplets P, Q, R, conformal projections on A_{3}^{1}(M), affine transformations of operators from Span_{F(M)}{I_{1}, I_{7}, I_{8}, G_{1}, ..., G_{4}} such that P + Q + R = I_{1}, PQ = QP = RQ = QR = 0, which produce the splitting A_{3}^{1}(M) = ImP ⊕ ImQ ⊕ ImR. Theorem 2.3.2 Let (M,g) be a Riemann space.

We consider P = x_{1}I_{1} + x_{2}I_{2} + x_{6}I_{6} + x_{7}G_{1} a conformal projection on A_{2}^{1}(M),

T = y_{1}I_{1} + y_{7}I_{7} + y_{8}I_{8} + ∑_{i=1}^{4}z_{i}G_{i}, T′ = y′_{1}I_{1} + y′_{7}I_{7} + y′_{8}I_{8} + ∑_{i=1}^{4}z′_{i}G_{i} conformal

projections on A_{3}^{1}(M) and R = I_{1} − I_{2} affine transformation on A_{3}^{1}(M). Let

(A^{∘}, A) be a pair from A_{3}^{1}(M) associated by the rule A_{stl}^{r} = \frac{∂Γ_{sl}^{r}}{∂x^{t}} + Γ_{mt}^{r}Γ_{st}^{m} to the

pair (Γ^{∘}, Γ) from A_{2}^{1}(M), where Γ^{∘} is the Levi-Civita connection corresponding to g,

and P Γ^{∘}= Γ,

(R^{∘}, R) be a pair from A_{3}^{1}(M) containing the curvature tensor field associated to Γ^{∘}

and the exotic curvature tensor field associated to P Γ^{∘}= Γ, C be the exotic conformal Weyl curvature tensor field. The diagrame

(∗)  \begin{aligned}Γ^{∘} & \underrightarrow{P} & Γ \\ ↓ & & ↓ \\ A^{∘} & & A \\ R ↓ & & ↓ R \\ R^{∘} & & R \\ T′ ↘ & & ↙ T \\ & C \end{aligned}

is closed for the next cases

a) P = I_{1} − \frac{1}{n}I_{2}, T = T′ arbitrary conformal projections on A_{3}^{1}(M);

b) P = \frac{1}{n}I_{2}, T′ = 0, T arbitrary conformal projections on A_{3}^{1}(M);

c) P = I_{1} + x_{2}I_{2} − (1 + nx_{2})I_{6}, x_{2} ≠ −\frac{1}{n}, T = T′ = 0 or T = T′ = I_{1} − \frac{1}{n − 1}I_{7} +

\frac{1}{n − 1}I_{8};

d) P = x_{2}I_{2} + (1 − nx_{2})I_{6}, x_{2} ≠ \frac{1}{n}, T = T′ = 0 or T′ = 0 and T = I_{1} − \frac{1}{n − 1}I_{7} +

\frac{1}{n − 1}I_{8};

e) P = I_{1} − \frac{1}{n}I_{2} + x_{6}I_{6} − x_{6}G_{1}, T = T′ = I_{1} − \frac{1}{n − 2}I_{7} + \frac{1}{n − 2}I_{8}+

+\frac{1}{n − 2}G_{1} − \frac{1}{n − 2}G_{2} + \frac{1}{(n − 1)(n − 2)}G_{3} − \frac{1}{(n − 1)(n − 2)}G_{4}.

Moreover, the diagrame reflects an invariance of gauge type of the exotic conformal Thomas connection and the exotic conformal Weyl curvature tensor field with respect to the orthogonal group. Proposition 2.3.1 The conformal Weyl curvature tensor field is invariant to in-

finitely many transformations of exotic conformal connections Γ^{∘} \underrightarrow{P}Γ, P = I_{1} −

\frac{1}{n}I_{2} + x_{6}I_{6} − x_{6}G_{1}, x_{6} ∈ F(M).

For x_{6} = −\frac{1}{n} one finds P Γ^{∘}= Γ, the conformal Thomas connection. This extension of the known properties of the conformal Weyl curvature tensor illustrates strikingly the generality of our method.

The study of the invariants and the geometrical interpretation of the results for the general diagram (∗), built with operators generated by all elements of the bases from CInv(3) and CInv(4) remain an open problem.

3 Pairs of geometrical object fields via H- projective invariant operators

The decomposition problem in holomorphically projective geometry was studied in [29]. We continue the research in this direction introducing the algebra of H-projective invariant operators.

3.1 Algebra HInv(r) of H-projective invariant operators

Let V be a real n-dimensional vector space, n = 2m and T_{r}^{r}(V) the vector space of tensors of type (r, r) on V. Using the Kronecker symbol δ = (δ_{j}^{i}) and the complex structure J = (J_{j} ^{i}) of V we define the vector subspace

HInv(r) = { ∑_{σ∈S_{r}} x_{σ}δ_{j_{1}}^{i_{σ(1)}} … δ_{j_{r}}^{i_{σ(r)}}+

+ ∑_{τ(1)<τ(2)} x_{σ τ} J_{j_{τ(1)}} ^{i_{σ(1)}}J_{j_{τ(2)}} ^{i_{σ(2)}}δ_{j_{τ(3)}}^{i_{σ(3)}} … δ_{jτ(r)}^{i_{σ(r)}} | x_{σ}, x_{στ} ∈ R, σ, τ ∈ S_{r}} ,

where S_{r} is the group of permutations ( H is the abbreviation from ”holomorphic”).

Every element P ∈ HInv(r) is an invariant tensor with respect to the subgroup G = {A ∈ GL(V)|AJ = JA} (i.e. ∀A ∈ G, A ∘ P = P ). We interpret every tensor of HInv(r) like an endomorfism on T_{r−1}^{1}(V), so HInv(r) is a real algebra the algebra of absolut invariant tensors Inv(r) being a subalgebra.

Every P = {P_{a}^{bc} _{st}^{r}} ∈ HInv(3) is an endomorphism on T_{2}^{1}(V), having the simplified expression P = ∑_{i=1}^{6} x_{i}I_{i} + ∑_{j=1}^{18} x_{j+6}H_{j}, where {I_{1}, …, I_{6}, H_{1}, …, H_{18}} is the basis in HInv(3).

For r = 4, every element of HInv(4) is written in the basis {I_{1}, …, I_{24}, H_{1}, …, H_{144}} like P = ∑_{i=1}^{24} y_{i}I_{i} + ∑_{j=1}^{144} v_{j}H_{j}. If the Ricci tensor is symmetric, then an operator P ∈ Span{I_{1}, I_{7}, I_{8}, H_{1}, …, H_{8}} produces the H - proiective curvature tensor field. These operators appear during the study of some close diagrame, which reflects an invariance of gauge type of the exotic H-proiective J-connections and the exotic H-proiective curvature tensor fields.

For geometrical reasons we study in detail a subalgebra in HInv(4), namely E = Span{I_{1}, I_{7}, I_{8}, H_{1}, H_{3}, H_{5}, H_{6}, H_{7}, H_{8}}. Indeed, the K - curvature tensor fields are generated by certain operators from the subalgebra E. If P ∈ Span{I_{1}, I_{7}, I_{8}, H_{1}, …, H_{8}} is a H - proiective projection, then P ∈ E.

Endowing E with a structure of Lie algebra, all the subalgebras are Lie subalgebras. It is interesting to analyse the infinite family of H-proiective projections, in par-

ticular of E and E, where E = Span{I_{1}, I_{2}, I_{6}, H_{1}, H_{2}}, which aroses in the study of some geometrical object fields. Indeed, the affine transformation of a projection from HInv(3) produces a symmetric H-projective J - connection. Moreover, the space of tensors of K - curvature type is connected with the subset of projections of the subalgebra E.

3.2 δ-J-decompositions of geometrical object fields

Let (M, J) be an almost complex n-dimensional manifold (n = 2m) and T_{r−1}^{1}(M) the F(M)-module of tensor fields of type (1, r − 1). Any H-proiective operator P on T_{r−1}^{1}(M) is extended to T_{r−1}^{1}(M), imposing the condition x_{σ}, x_{στ} ∈ F(M), σ, τ ∈ S_{r}, and generates a F(M)-module H-proiective invariant tensor fields, denoted HInv(r).

Let A_{r−1}^{1}(M) be the affine F(M)-module over T_{r−1}^{1}(M), built in the subsection 1.3. We denote also with P the affine transformation induced on the affine F(M)- module A_{r−1}^{1}(M).

If Γ ∈ C, the space of affine connections, one gets the pair of geometrical object fields (Γ, P Γ) from A_{2}^{1}(M). In the set of J-connections we determine properties of J-connections compatible with a Hermitian metric and in particular we study the case of the Kähler manifolds. Theorem 3.2.1 Let (M, g, J) be a Kähler manifold. The H-proiective projections

P ∈ E which produce pairs of geometrical object fields (Γ^{∘}, Γ = P Γ^{∘}), where Γ^{∘} is the Levi-Civita connection and Γ a J-connection are

P = I_{1}, P = I_{1} + x_{2}I_{2} + x_{6}I_{6} + (1 + nx_{2} + x_{6})H_{1} − x_{2}H_{2} ,

P = 0, P = x_{2}I_{2} + x_{6}I_{6} + (1 − nx_{2} − x_{6})H_{1} − x_{2}H_{2}, x_{2}, x_{6} ∈ F(M) . Remark 3.2.1 In the particular case x_{2} = x_{6} = −x_{7} = −x_{8} = −\frac{1}{n + 2} , x_{1} = 1

one gets the pair of connections (Γ^{∘}, P Γ^{∘}= Γ), where Γ is a J-connection H-projective related with the Levi-Civita connection and P = I_{1} − \frac{1}{n + 2}(I_{2} + I_{6} − H_{7} − H_{8}) is

the unique H-projective projection which applies Γ^{∘} to the symmetric J-connection

Γ = P Γ^{∘}.

Let (M, g, J) be a Kähler manifold and L_{K}(M) the F(M)-module of K-curvature tensor fields i.e. the set of tensor fields R_{jkl}^{i} of type (1, 3) verifying

1) R_{jkl}^{i} + R_{jlk}^{i} = 0, 2) R_{ijkl} + R_{jikl} = 0, where R_{ijkl} = g_{si}R_{jkl}^{s},

3) ∑_{j,k,l}^{c}R_{jkl}^{i} = 0 (Bianchi I), 4) R_{stl}^{r}J_{r} ^{p} = J_{s} ^{r}R_{rtl}^{p} (Kähler identity).

Theorem 3.2.2 There are infinitely many endomorphisms

P ∈ Span_{F(M)}{I_{1}, I_{7}, I_{8}, H_{1}, …, H_{8}} in HInv(4), such that P apply L_{K}(M) into L_{K}(M). Corollary 3.2.1 The H-projective projections P ∈ E defined on L_{K}(M) are P = I_{1} + y_{7}I_{7} + y_{8}I_{8} + v_{1}H_{1} + v_{3}H_{3} + v_{3}H_{5} + v_{1}H_{6} − y_{8}H_{7} − y_{7}H_{8}, where

a) \frac{1}{2} + ny_{7} + y_{8} − v_{1} = 0, v_{3} arbitrary; b) y_{7} = y_{8} = v_{1} = v_{3} = 0 and the supplementary projections Q = y_{7}I_{7} + y_{8}I_{8} + v_{1}H_{1} + v_{3}H_{3} + v_{3}H_{5} + v_{1}H_{6} − y_{8}H_{7} − y_{7}H_{8}, where

a) ny_{7} +y_{8} −v_{1} = \frac{1}{2}, v_{3} arbitrary; b) y_{7} = y_{8} = v_{1} = v_{3} = 0. Moreover, PR = y_{1}R and these operators are symmetric with respect to the induced inner product on L_{K}(M). Also, for every projection, one gets L_{K}(M) = ImP, ImQ = 0. Remark 3.2.2 If

P = I_{1} − \frac{n}{n^{2} − 4}I_{7} + \frac{n}{n^{2} − 4}I_{8} − \frac{n}{n^{2} − 4}H_{1} − \frac{1}{n + 2}H_{2} + \frac{n}{n^{2} − 4}H_{3}+

+\frac{1}{n + 2}H_{4} + \frac{2}{n^{2} − 4}H_{5} − \frac{2}{n^{2} − 4}H_{6} − \frac{2}{n^{2} − 4}H_{7} + \frac{2}{n^{2} − 4}H_{8}

and R ∈ L′_{K}(M), then PR is the H-proiective curvature tensor field and PR ∈ L′_{K}(M), the set of (1,3)-tensor fields verifying 1), 3), 4).

Hence L′_{K}(M) = ImP ⊕ KerP, where ImP = {L ∈ L′_{K}(M) with vanishing Ricci tensor field }.

We determine some pairs of geometrical object fields from A_{2}^{1}(M), respectively A_{3}^{1}(M) associated by affine transformations of certain H-projective operators from HInv(3), HInv(4), generalizing properties of J-connections H-projective related with the Levi-Civita connection and of the H-projective curvature tensor field. Theorem 3.2.3 Let (M, g, J) be a Kähler manifold and

P = x_{1}I_{1} + x_{2}I_{2} + x_{6}I_{6} + x_{7}H_{1} − x_{2}H_{2}, P = y_{1}I_{1} + y_{7}I_{7} + y_{8}I_{8} + v_{1}H_{1}+ +v_{2}H_{2} + v_{3}H_{3} + v_{4}H_{4} + v_{5}H_{5} + v_{6}H_{6} + v_{7}H_{7} + v_{8}H_{8} , T = z_{1}I_{1} + z_{7}I_{7} + z_{8}I_{8} + w_{1}H_{1} + w_{2}H_{2} + w_{3}H_{3} + w_{4}H_{4} + w_{5}H_{5} + w_{6}H_{6}+ +w_{7}H_{7} + w_{8}H_{8}, R = I_{1} − I_{2}

affine transformations on A_{2}^{1}(M), respectively A_{3}^{1}(M).

There are infinitely many H-projective projections P and H-projective transformations P, T such that next diagrame

(∗)  \begin{aligned}Γ^{∘} & \underrightarrow{P} & Γ \\ ↓ & & ↓ \\ A^{∘} & & A \\ R ↓ & & ↓ R \\ R^{∘} & & R \\ T ↘ & & ↙P \\ & H \end{aligned}

is closed where (Γ^{∘}, Γ = P Γ^{∘}) is a pair from A_{2}^{1}(M), Γ^{∘} is the Levi-Civita connection and Γ the exotic J-connection H-projective related with Γ^{∘},

(A^{∘}, A) is a pair from A_{3}^{1}(M), associated to (Γ^{∘}, Γ), by the rule A_{stl}^{r} = \frac{∂Γ_{sl}^{r}}{∂x^{t}} + Γ_{mt}^{r}Γ_{sl}^{m},

(R^{∘}, R) is a pair from A_{3}^{1}(M), R^{∘} being the curvature tensor field associated to Γ^{∘}, R the exotic curvature tensor field associated to Γ, H the exotic H-projective curvature tensor field.

In the particular case

P = I_{1} − \frac{n}{n^{2} − 4}I_{7} + \frac{n}{n^{2} − 4}I_{8} − \frac{n}{n^{2} − 4}H_{1} − \frac{1}{n + 2}H_{2} + \frac{n}{n^{2} − 4}H_{3}+

+\frac{1}{n + 2}H_{4} + \frac{2}{n^{2} − 4}H_{5} − \frac{2}{n^{2} − 4}H_{6} − \frac{2}{n^{2} − 4}H_{7} + \frac{2}{n^{2} − 4}H_{8}.

one gets PR the H-projective curvature tensor field. The H-projective projection P = I_{1}− \frac{1}{n + 2}(I_{2}+I_{6}−H_{1}−H_{2}) produces the pair of connections (Γ^{∘}, P Γ^{∘}= Γ), where

Γ is the J-connection H-projective related with Γ^{∘}. If T = P we find again that the Hprojective curvature tensor field is invariant under the H-projective transformations of J-connections.

It remains an open problem the geometric interpretation and the study of the generale diagrame (∗) built with operators generated by all elements of the bases of HInv(3) and HInv(4).

4 Pairs of geometrical object fields via conformal holomorphic invariant operators

The complete decomposition of K-curvature tensors [32], [26], [19] characterizes spaces of constant holomorphic sectional curvature. Introducing the algebra CHInv(r) of δ − J − g operators, we extend the Sitaramayya-Mori decomposition and also we determine general invariants under some transformations of geometrical object fields.

4.1 Algebra CHInv(r) of conformal holomorphic invariant operators

Let V be a real n-dimensional vector space, n = 2m, endowed with the complexe structure J = (J_{j} ^{i}) and the Hermitian inner product g = (g_{ij}), g^{−1} = (g^{ij}). Let δ = (δ_{j}^{i}) be the Kronecker symbol, J_{ij} = J_{i} ^{k}g_{kj} and T_{r}^{r}(V) the vector space of tensors of type (r, r) on V.

The triplet (δ, J, g) determines the next vector subspace

CHInv(r) = {P_{j_{1}...j_{r}}^{i_{1}...i_{r}} = ∑_{σ∈S_{r}} x_{σ}δ_{j_{1}}^{i_{σ(1)}}...δ_{j_{r}}^{i_{σ(r)}}+

+ ∑_{σ,τ∈S_{r}, τ(1)<τ(2), σ(1)<σ(2)} x_{στ} g_{j_{τ(1)}j_{τ(2)}}g^{i_{σ(1)}i_{σ(2)}}δ_{j_{τ(3)}}^{i_{σ(3)}}...δ_{j_{τ(r)}}^{i_{σ(r)}}+

+ ∑_{σ,τ∈S_{r}, τ(1)<τ(2)} x_{στ} J_{j_{τ(1)}} ^{i_{σ(1)}}J_{j_{τ(2)}} ^{i_{σ(2)}}δ_{j_{τ(3)}}^{i_{σ(3)}}...δ_{j_{τ(r)}}^{i_{σ(r)}}+

+ ∑_{σ,τ∈S_{r}, σ(1)<σ(2), τ(1)<τ(2)} x_{στ} J_{j_{τ(1)}j_{τ(2)}}J^{i_{σ(1)}i_{σ(2)}}δ_{j_{τ(3)}}^{i_{σ(3)}}...δ_{j_{τ(r)}}^{i_{σ(r)}}+

+ ∑_{σ,τ∈S_{r}, τ(1)<τ(2), σ(2)<σ(3)} x_{στ} J_{j_{τ(1)}j_{τ(2)}}J_{j_{τ(3)}} ^{i_{σ(1)}}g^{i_{σ(2)}i_{σ(3)}}δ_{j_{τ(4)}}^{i_{σ(4)}}...δ_{j_{τ(r)}}^{i_{σ(r)}}+

+ ∑_{σ,τ∈S_{r}, σ(1)<σ(2), τ(2)<τ(3)} J^{i_{σ(1)}i_{σ(2)}}J_{j_{τ(1)}} ^{i_{σ(3)}}g_{j_{τ(2)}j_{τ(3)}}δ_{j_{τ(4)}}^{i_{σ(4)}}...δ_{j_{τ(r)}}^{i_{σ(r)}} |

x_{σ}, x_{στ} ∈ R, σ, τ ∈ S_{r}} , where J_{ij} = J_{i} ^{k}g_{kj}, J^{ij} = J_{k} ^{i}g^{kj}, S_{r} is the group of permutations and CH is the abbreviation of ”conformal holomorphic”. The subspace CHInv(r) is endowed with a structure of Lie algebra, considering any P as an endomorphism on T_{r−1}^{1}(V).

Any element P ∈ CHInv(r) is invariant with respect to the group Ḡ, where Ḡ = CO(V) ∩ Sp(m, V) ∩ G, G = {A ∈ GL(V)|AJ = JA}; ∀A ∈ Ḡ, A ∘ P = P.

We should remark that CHInv(r) contains the subalgebras Inv(r), CInv(r) and HInv(r). Hence the δ-operators, the δ-g-operators and the δ-J-operators belong to a much more general algebra of δ-J-g-operators.

For r = 3, the algebra CHInv(3) is generated by the endomorphisms P = ∑_{i=1}^{6} x_{i}I_{i} + ∑_{j=1}^{9} x_{j+6}G_{j} + ∑_{k=1}^{18} x_{15+k}H_{k} + ∑_{l=1}^{9} x_{33+l}C_{l} + ∑_{s=1}^{9} x_{42+s}B_{s} + ∑_{r=1}^{9} x_{51+r}K_{r}.

For geometrical reasons one studies the subalgebra F = Span{I_{1}, I_{2}, I_{6}, G_{1}, H_{1}, H_{2}, C_{1}}, which aroses during the developement of the theory of the conformal complex connections on Kähler manifolds [45]. Introducing a structure of Lie algebra on F, all its subspaces are Lie subalgebras.

In the case r = 4, any element P = {P_{a}^{bcd} _{stl}^{r}} of CHInv(4), has the simplified expression P = ∑_{i=1}^{24} y_{i}I_{i} + ∑_{j=1}^{72} z_{j}G_{j} + ∑_{k=1}^{144} v_{r}H_{r} + ∑_{l=1}^{72} u_{l}C_{l} + ∑_{s=1}^{72} w_{s}B_{s} + ∑_{p=1}^{72} x_{p}K_{p}, where {I_{1}, ..., I_{24}, G_{1}, ..., G_{72}, H_{1}, ..., H_{144}, C_{1}, ..., C_{72}, B_{1}, ..., B_{72}, K_{1}, ..., K_{72}} is the basis in CHInv(4).

The subset F = Span{I_{1}, I_{7}, I_{8}, G_{1}, …, G_{4}, B_{1}, …, B_{6}, H_{5}, H_{6}, H_{11}}, which contains the algebra F′ = Span{I_{1}, I_{7}, I_{8}, G_{1}, …, G_{4}, B_{1}, B_{2}, B_{3}}, appears in the generalization of the theory of the Bochner curvature tensors on almost Hermitian manifolds. In the infinite set of conformal holomorphic projections one projection from CHnv(4) produces the Bochner curvature tensor field and the affine transformation of a projection from CHnv(3) a complex conformal connection in the case of a Kähler manifold. Remark 4.1.1 In particular, if P ∈ F′ is a conformal holomorphic projection, for w_{1} = w_{2} = w_{3} = 0, one gets a conformal projection and for z_{1} = z_{2} = z_{3} = z_{4} = 0 one finds a projective projection. Theorem 4.1.1 There are infinitely many conformal holomorphic nonvanishing projections P, Q, R ∈ F′ such that P + Q + R = I_{1}, PQ = QP = RQ = QR = 0 and T_{3}^{1}(V) = ImP ⊕ ImQ ⊕ ImR.

4.2 δ-J-g-decompositions of geometrical object fields

Let (M, g, J) be an almost Hermitian manifold, T_{r−1}^{1}(M) the F(M)-module of tensor fields of type (1, r − 1) and A_{r−1}^{1}(M) the affine F(M)-module over T_{r−1}^{1}(M), built in the subsection 1.3.

The conformal holomorphic operators P on T_{r−1}^{1}(M) are extended to T_{r−1}^{1}(M) imposing x_{σ}, x_{τσ} ∈ F(M), σ, τ ∈ S_{r} and generate a F(M)-module of invariant endomorphisms denoted CHInv(r). The affine transformation of P induced on the F(M)–module A_{r−1}^{1}(M), denoted also with P, produces a conformal holomorphic decomposition of the space A_{r−1}^{1}(M). Theorem 4.2.1 Let (M, g, J) be a Kähler manifold. The conformal holomorphic projections of F = Span_{F(M)}{I_{1}, I_{2}, I_{6}, G_{1}, H_{1}, H_{2}, C_{1}} which produce pairs of geometri-

cal object fields (Γ^{∘}, Γ = P Γ^{∘}), where Γ^{∘} is the Levi–Civita connection and Γ = P Γ^{∘} is a J-connection, called the exotic complex conformal connection, are P = I_{1}, P = I_{1} + x_{2}I_{2} + x_{6}I_{6} + x_{7}(G_{1} + C_{1}) + (1 + nx_{2} + x_{6} + 2x_{7})H_{1} − x_{2}H_{2}, P = 0, P = x_{2}I_{2} + x_{6}I_{6} + x_{7}(G_{1} + C_{1}) + (1 − nx_{2} − x_{6} − 2x_{7})H_{1} − x_{2}H_{2}. Remark 4.2.1 In the particular case x_{2} = x_{6} = x_{34} = −x_{7} = −x_{16} = −x_{17} =

−\frac{1}{n + 2}, x_{1} = 1 one finds P = I_{1} − \frac{1}{n + 2}(I_{2} + I_{6} − G_{1} − H_{1} − H_{2} + C_{1}) and the pair

of connections (Γ^{∘}, P Γ^{∘}), where P Γ^{∘} is a complex conformal connection on a Kähler manifold (M, g, J) [45].

For r = 4, let (M, g, J) be an almost Hermitian manifold and let F be the F(M)- module generated by the operators P = y_{1}I_{1} + y_{7}I_{7} + y_{8}I_{8} + ∑_{i=1}^{4} z_{i}G_{i} + ∑_{j=1}^{6} w_{j}B_{j} + v_{5}H_{5} + v_{6}H_{6} + v_{11}H_{11}.

Let R ∈ L_{K}(M), L_{K}(M) being the space of K–curvature tensor fields on M. We define R̄ = PR, the exotic Bochner curvature tensor field and we study the pair of geometrical object fields (R, PR). Theorem 4.2.2 The endomorphisms P of F applying the space L_{K}(M) into L_{K}(M) are

P = y_{1}I_{1} + y_{7}(I_{7} − I_{8} − G_{1} + G_{2} + B_{4} − B_{5} + H_{5} − H_{6} + 2H_{11} + 2B_{6})+ +z_{3}(G_{3} − G_{4} + B_{1} − B_{2} + 2B_{3}),

where y_{1}, y_{7}, z_{3} ∈ F(M). Remark 4.2.2 Our general approuch is motivated by the particular case y_{7} =

−\frac{1}{n + 4}, z_{3} = \frac{1}{(n + 1)(n + 4)} which produces the pair of tensor fields (R, PR), R

being the curvature tensor field and PR = R̄ the Bochner curvature tensor field defined for the class of almost Hermitian manifolds. Theorem 4.2.3 Let (M, g, J) be an almost Hermitian manifold and

P =∑_{i=1}^{6} x_{i}I_{i} + ∑_{j=1}^{9} x_{j+6}G_{j} + ∑_{k=1}^{18} x_{15+k}H_{k} + ∑_{l=1}^{9} x_{33+l}C_{l} + + ∑_{s=1}^{9} x_{42+s}B_{s} + ∑_{r=1}^{9} x_{51+r}K_{r},

P = y_{1}I_{1} + y_{7}I_{7} + y_{8}I_{8} + ∑_{i=1}^{4} z_{i}G_{i} + ∑_{j=1}^{6} w_{j}B_{j} + v_{5}H_{5} + v_{6}H_{6} + v_{11}H_{11}, P′ = y′_{1}I_{1} + y′_{7}I_{7} + y′_{8}I_{8} + ∑_{i=1}^{4} z′_{i}G_{i} + ∑_{j=1}^{6} w′_{j}B_{j} + v′_{5}H_{5} + v′_{6}H_{6} + v′_{11}H_{11}, T = I_{1} − I_{2} affine transformations on A_{2}^{1}(M) and A_{3}^{1}(M). There are infinitely many transformations P, P and P′ such that

Γ^{∘}  \underrightarrow{P}  Γ
↓  ↓
A^{∘}  A
T ↓  ↓ T
R^{∘}  R
P |_{⟶} B _{⟵} | P′

is a closed diagrame, where (Γ^{∘}, Γ) is a pair from A_{2}^{1}(M), Γ^{∘} being the Levi-Civita connection and Γ = P Γ^{∘},

(A, A^{∘}) is a pair from A_{3}^{1}(M) associated to (Γ, Γ^{∘}), by the rule A_{stl}^{r} = \frac{∂Γ_{sl}^{r}}{∂x^{t}} +Γ_{mt}^{r}Γ_{sl}^{m},

(R, R^{∘}) the exotic curvature tensor fields associated to (Γ, Γ^{∘}), R = T A, R^{∘}= T A^{∘} and

B the exotic Bochner curvature tensor B = P′R = P R^{∘}.

This is a general method to determine the transformations of exotic connections under which the Bochner curvature tensor field is invariant.

4.3 Extension of the Sitaramayya - Mori decomposition

In [26] H.Mori studied the decomposition of the space of K-curvature tensors. The same orthogonal decomposition, for a real Hermitian n-dimensional vector space, V, n = 2m, is determined by M.Sitaramayya [32] : L_{K}(V) = L_{K}^{1}(V) ⊕ L_{K}^{W} (V) ⊕ L_{K}^{2}(V), where

L_{K}^{1}(V) = {L ∈ L_{K}(V) | L with constant holomorphic sectional curvature }, L_{K}^{W} (V) = {L ∈ L_{K}(V) | L with vanishing Ricci tensor }, L_{K}^{W} (V) ⊕ L_{K}^{2}(V) = {L ∈ L_{K}(V) | T r(L) = 0}, L_{K}^{1}(V) ⊕ L_{K}^{W} (V) = {L ∈ L_{K}(V) | K(L) = λI, λ ∈ R}. The theory of conformal holomorphic operators gives a new method to get the

Sitaramayya-Mori decomposition. Studing the nonvanishing conformal holomorphic operators from F, one finds the orthogonal decomposition. Theorem 4.3.1 The conformal holomorphic projections P, Q, R ∈ F, defined on L_{K}(M) such that P + Q + R = I_{1}, PQ = QP = 0, RQ = QR = 0 and L_{K}(M) = ImP ⊕ ImQ ⊕ ImR are:

P = \frac{1}{n + 2}(H_{3} − H_{4} + B_{1} − B_{2} + 2B_{3}),

\begin{aligned}Q & = & I_{1} − \frac{1}{n + 4} \{[I_{7} − I_{8} − H_{1} + H_{2} + B_{4} − B_{5} + H_{5} − H_{6} + 2H_{11}] − \\ & − & \frac{1}{n + 2}[H_{3} − H_{4} + B_{1} + 2B_{3}]\}, \end{aligned}

\begin{aligned}R & = & \frac{1}{n + 4} \{[I_{7} − I_{8} + H_{1} − H_{2} + B_{4} − B_{5} + H_{5} − H_{6} + 2H_{11}] − \\ & − & \frac{n}{2}[H_{3} − H_{4} + B_{1} − B_{2} + 2B_{3}]\}. \end{aligned}

Using general affine transformations of conformal holomorphic projections on A_{3}^{1}(M) it is possible to characterize all splittings into three components of this affine space. Theorem 4.3.2 There are infinitely many nonvanishing triplets P, Q, R, conformal holomorphic projections on A_{3}^{1}(M), affine transformations of operators from Span_{F(M)}{I_{1}, I_{7}, I_{8}, G_{1}, ..., G_{4}, B_{1}, B_{2}, B_{3}}, such that P + Q + R = I_{1}, PQ = QP = 0, RQ = QR = 0. Moreover A_{3}^{1}(M) = ImP ⊕ ImQ ⊕ ImR,

P = I_{1} + y_{7}I_{7} + y_{8}I_{8} + ∑_{i=1}^{4} z_{i}G_{i} + ∑_{i=1}^{3} w_{i}B_{i},

Q = −y_{7}I_{7} − y_{8}I_{8} − z_{1}G_{1} − z_{2}G_{2} + \frac{y_{7} + z_{2}}{n} G_{3} + \frac{y_{8} + z_{1}}{n} G_{4},

R = −(z_{3} + \frac{y_{7} + z_{2}}{n} )G_{3} − (z_{4} + \frac{y_{8} + z_{1}}{n} )G_{4} − ∑_{i=1}^{3} w_{i}B_{i},

where 1 + ny_{7} + y_{8} + z_{1} = 0, z_{2} + nz_{3} + z_{4} + w_{1} + w_{3} = 0, w_{2} being arbitrary.

The F(M)-module of invariant (r, r)-tensor fields Inv(r), CInv(r), HInv(r), CHInv(r) were built interpreting every element as an operator on T_{r−1}^{1}(M), which has affine induced transformations on the affine F(M) - modulule A_{r−1}^{1}(M) of geometrical object fields of type (1, r − 1). It is possible to built another algebras considering any operator as an endomorphism on T_{r}^{0}(M) (the multiplication P Q being given by the rule P_{j_{1}...j_{r}}^{i_{1}...i_{r}}Q_{k_{1}...k_{r}}^{j_{1}...j_{r}}), having an affine transformation on an affine F(M)-module A_{r}^{0}(M) of geometrical object fields of type (0, r), defined similar with A_{r−1}^{1}(M).

We should remark that it is worthwhile to study also the cases r = 2 or r = 6. Indeed R·R, Q(g, R), the tensor fields which characterize the pseudo-symmetry or the semi-symmetry property are produced by affine transformations on A_{6}^{0}(M) of some absolut invariant operators of Inv(6). Also, the Obata operators Ω and Ω′, which determine the pairs of linear connections compatible with certain G-structures are built by some affine transformations on A_{2}^{0}(M) of conformal invariant operators of CInv(2).

Particular projective projections, defined using the representation theory, were studied in [22], illustrating that there is a connection between the subgroups of the permutation group S_{r} and the corresponding subalgebras of operators, built using our theory.

5 Geometry of pairs of connections

5.1 Geodesically and subgeodesically related manifolds

Spaces with constant curvature and Einstein spaces geodesically related were studied in [43]. Properties of recurrent and birecurrent spaces geodesically related were obtained in [41], [28]. In [5] appeared another direction of study on Riemannian manifolds. Pseudo-symmetric spaces, which verify

(∗)R · R and Q(g, R) are linear dependent at every point of M, represent a natural generalization of the semi-symmetric manifolds (R · R = 0) [34] and arrose during the study of the totally umbilical submanifolds of semi-symmetric spaces, as well as during the consideration of geodesic mappings. The condition (∗) is equivalent to

R · R = LQ(g, R) , L being defined on the set U = {x ∈ M | R ≠ R(1) at x},

R(1) = \frac{1}{n(n − 1)}(G_{3} − G_{4})R,

(R · R)_{hijklm} = (δ_{hijklm}^{adefbc} − δ_{hijklm}^{daefbc} − δ_{hijklm}^{efadbc} − δ_{hijklm}^{efdabc})(C_{4}^{1}(R ⊗ R))_{abcdef},

C_{4}^{1} being the contraction of the (1,7)-tensor R ⊗ R, (Q(g, R))_{hijklm} = (δ_{hijklm}^{adefcb} − δ_{hijklm}^{bdcfea} − δ_{hijklm}^{caefdb} − δ_{hijklm}^{caefbd}−

−δ_{hijklm}^{cdafeb} − δ_{hijklm}^{cdafbe} − δ_{hijklm}^{cdeafb} − δ_{hijklm}^{cdeabf})(g ⊗ R)_{abcdef}.

Replacing the Riemann tensor with the conharmonic tensor,

Z = [I_{1} − \frac{1}{n − 2}(I_{7} − I_{8} − G_{1} + G_{2})]R, one gets the tensor R · Z. If R · Z = 0, then

(M, g) is a conharmonic semi-symmetric space (or Z-semi-symmetric).

For A, D symmetric tensors of type (0, 2), we define

(R · A)_{hijk} = (−δ_{hijk}^{abcd} + δ_{hijk}^{abdc})(C_{1}^{1}(A ⊗ R))_{abcd},

Q(g, D) = (δ_{hijk}^{abcd} − δ_{hijk}^{acbd} − δ_{hijk}^{cadb} − δ_{hijk}

6 Geometry of pairs of connections

6.1 Geodesically and subgeodesically related manifolds

Spaces with constant curvature and Einstein spaces geodesically related were studied in [43]. Properties of recurrent and birecurrent spaces geodesically related were obtained in [41], [28]. In [5] appeared another direction of study on Riemannian manifolds. Pseudo-symmetric spaces, which verify

(∗)R · R and Q(g, R) are linear dependent at every point of M, represent a natural generalization of the semi-symmetric manifolds (R · R = 0) [34] and arrose during the study of the totally umbilical submanifolds of semi-symmetric spaces, as well as during the consideration of geodesic mappings. The condition (∗) is equivalent to

R · R = LQ(g, R) , L being defined on the set U = {x ∈ M | R ≠ R(1) at x},

R(1) = \frac{1}{n(n − 1)}(G_{3} − G_{4})R,

(R · R)_{hijklm} = (δ_{hijklm}^{adefbc} − δ_{hijklm}^{daefbc} − δ_{hijklm}^{efadbc} − δ_{hijklm}^{efdabc})(C_{4}^{1}(R ⊗ R))_{abcdef}, C_{4}^{1} being the contraction of the (1,7)-tensor R ⊗ R,

(Q(g, R))_{hijklm} = (δ_{hijklm}^{adefcb} − δ_{hijklm}^{bdcfea} − δ_{hijklm}^{caefdb} − δ_{hijklm}^{caefbd}−

−δ_{hijklm}^{cdafeb} − δ_{hijklm}^{cdafbe} − δ_{hijklm}^{cdeafb} − δ_{hijklm}^{cdeabf})(g ⊗ R)_{abcdef}.

Replacing the Riemann tensor with the conharmonic tensor,

Z = [I_{1} − \frac{1}{n − 2}(I_{7} − I_{8} − G_{1} + G_{2})]R, one gets the tensor R · Z. If R · Z = 0, then

(M, g) is a conharmonic semi-symmetric space (or Z-semi-symmetric).

For A, D symmetric tensors of type (0, 2), we define

(R · A)_{hijk} = (−δ_{hijk}^{abcd} + δ_{hijk}^{abdc})(C_{1}^{1}(A ⊗ R))_{abcd},

Q(g, D) = (δ_{hijk}^{abcd} − δ_{hijk}^{acbd} − δ_{hijk}^{cadb} − δ_{hijk}^{cabd})(g ⊗ D)_{abcd}, where C_{1}^{1} is the contraction and ”,” is the covariant derivative with respect to g.

Two Riemann spaces V_{n} = (M, g) and \overline{V_{n}} = (M, \overline{g}) are ξ^{i}-subgeodesically related, the tensor of correspondence being −g_{ij}, if the Yano formulae are satisfied

\overline{Γ}_{jk}^{i} = Γ_{jk}^{i} + δ_{j}^{i}ψ_{k} + δ_{k}^{i}ψ_{j} − g_{kj}ξ^{i}, where ψ_{i} şi ξ^{i} are the components of a 1-form, respectively vector field. This is equivalent with the existence of a difeomorphism f, called subgeodesic map, between these two spaces, which applies the ξ^{i}-subgeodesics into the ξ^{i}-subgeodesics.

V_{n} and \overline{V_{n}} are trivial subgeodesically related if Ψ_{i} − ξ_{i} = 0, ∀i ∈ \{1, ..., n\}.

If ξ^{i} = 0 then V_{n} and \overline{V_{n}} are geodesically related and the Yano formulae become the Weyl formulae. If Ψ_{ij} = fg_{ij}, where f ∈ F(M), then V_{n} and \overline{V_{n}} are special geodesically related. The correspondence is trivial if Ψ_{i} = 0, ∀i ∈ {1, ..., n}.

The Weyl projective curvature tensor field W = [I_{1} − \frac{1}{n − 1}(I_{7} −I_{8})]R is invariant

under geodesic maps.

We study Riemannian manifolds geodesically related with conharmonic semisymmetric spaces and also pseudo-symmetric manifolds subgeodesically related. Theorem 5.1.1 Let V_{n} = (M, g) and \overline{V_{n}} = (M, \overline{g}), n ≥ 3, be nontrivial geodesically related Riemannian manifolds. If \overline{V_{n}} is \overline{Z}-semi-symmetric, then V_{n} and \overline{V_{n}} are spaces with constant curvature or the geodesic correspondence is speciale.

Theorem 5.1.2 Let V_{n} = (M, g) and \overline{V_{n}} = (M, \overline{g}), n ≥ 3, be nontrivial geodesically related Riemannian manifolds. If \overline{V_{n}} is a space with irreducible curvature tensor field and \overline{V_{n}} is \overline{Z}-semi-symmetric, then V_{n} and \overline{V_{n}} are spaces with constant curvature. Theorem 5.1.3 Let V_{n} = (M, g) and \overline{V_{n}} = (M, \overline{g}), n ≥ 3, nontrivial ξ^{i}- subgeodesically related Riemannian manifolds, the tensor of correspondence being −g. If \overline{V_{n}} is a space with irreducible curvature tensor field and \overline{V_{n}} is \overline{Z}-semi-symmetric, then \overline{V_{n}} = (M, \overline{g}) and Ṽ_{n} = (M, g̃ = e^{2ξ}g) are spaces with constant curvature. Theorem 5.1.4 Let V_{n} = (M, g) and \overline{V_{n}} = (M, \overline{g}), n ≥ 3, be ξ^{i}-subgeodesica-lly related Riemannian manifolds, the tensor of correspondence being −g, such that ξ_{hk} = 0, where ξ_{hk} = ξ_{h,k} − ξ_{h}ξ_{k} + \frac{1}{2}ξ_{i}ξ^{i}g_{hk}. If \overline{V_{n}} = (M, \overline{g}) a conharmonic semi-symmetric space and with irreducible curvature tensor filed, then V_{n} = (M, g) is Einstein space. Theorem 5.1.5 Let V_{n} = (M, g) and \overline{V_{n}} = (M, \overline{g}), n ≥ 3, be two Riemann spaces ξ^{i}-subgeodesically related, the tensor of correspondence being −g, such that B = \frac{1}{n}T r(B)g, where B is the (0,2)–tensor field having the components B_{rs} = ξ_{r,s} − ξ_{r}ξ_{s}.

If V_{n} is a pseudo-symmetric space, then \overline{V_{n}} is a pseudo-symmetric space. Theorem 5.1.6 Let V_{n} = (M, g) and \overline{V_{n}} = (M, \overline{g}), n ≥ 3, be ξ^{i}-subgeodesically related Riemannian spaces, the tensor of correspondence being −g.

Then \overline{R} · g = Q(g, F), where F is the symmetric tensor of type (0, 2), having the components F_{ij} = ψ_{i;j} − ξ_{i;j} − (ψ_{i} − ξ_{i})(ψ_{j} − ξ_{j}), where ”;” represents the covariant derivatives with respect to the metric \overline{g}. Proposition 5.1.1 Let V_{n} = (M, g) and \overline{V_{n}} = (M, \overline{g}), n ≥ 3, be ξ^{i}-subgeodesically related Riemannian spaces, the tensor of correspondence being −g.

If \overline{V_{n}} is a pseudo-symmetric space, the map \overline{L} which satisfies (∗) on \overline{U} is constant and F_{ij} = fg_{ij} + h\overline{g}_{ij}, where f, h ∈ F(M), h being nonzero, then the relation

(\overline{L} + h) [g − \frac{1}{n}(\overline{g}^{ij}g_{ij})\overline{g}] = 0

holds on the set \overline{U}.

6.2 Semi-symmetric connections on Weyl generalized manifolds

Let (M, g) be a Riemannian space, n-dimensional, f ∈ F(M) be a nowhere vanishing function, ĝ = {e^{λ}g|λ ∈ F(M)} be the conformal structure generated by g and and the Weyl f-structure W^{f} : ĝ ↦ Λ^{1}(M), where W^{f}(e^{λ}g) = W^{f}(g) − fdλ, ∀λ ∈ F(M), Λ^{1}(M) being the space of the 1-forms on M. Weyl generalized manifolds (M, ĝ, W^{f}) are a natural extension of Weyl spaces (M, ĝ, W), obtained for f = 1 ([2]).

A linear connection ∇ on M is compatible with the Weyl f-structure W^{f} and associated to the 1-form ω if

(∇_{fX}g)(Y, Z) + W^{f}(g)(X)g(Y, Z) + ω(Y)g(X, Z) + ω(Z)g(X, Y) = 0.

∇ is called σ-semi-symmetric connection if T (X , Y) = σ(Y)X − σ(X)Y , where σ is a 1-form and T is the torsion tensor.

We study semi-symmetric connections on Weyl generalized manifolds. Theorem 5.2.1 Let (M , ĝ , W^{f}) be a Weyl generalized manifold. There exists a unique σ-semi-symmetric connection compatible with the Weyl f-stucture and associated to the 1-form ω, where g, f, W^{f}(g), ω are given, determined by

∇_{fX}Y =∇^{∘}_{fX} Y + \frac{1}{2}W^{f}(g)(X)Y + (\frac{1}{2}W^{f}(g) + fσ) (Y)X − g(X, Y)

(\frac{1}{2}W^{f}(g) + fσ − ω)^{#} ,

∇^{∘} being the Levi-Civita connection associated to g. Theorem 5.2.2 If n > 2, then the tensor field

D_{jkl}^{i} = R_{jkl}^{∗ i} − \frac{1}{n − 2} \{Ω_{jk}^{mi} (R_{ml}^{∗} − \frac{K}{2(n − 1)}g_{ml}) − Ω_{jl}^{mi} (R_{mk}^{∗} − \frac{K}{2(n − 1)}g_{mk})\} ,

where K is the scalar curvature, R_{jkl}^{∗ i} = R_{jkl}^{i}− \frac{1}{n}δ_{j}^{i}B_{hk}, B_{hk} = R_{shk}^{s}, R_{jk}^{∗} = R_{jki}^{∗ i} and

Ω = \frac{1}{2}(I ⊗I −g ⊗g̃), is an invariant under the transformation of a σ-semi-symmetric

connection, compatible with Weyl f-structure W^{f} into the \overline{σ}-semi-symmetric connection \overline{∇}, compatible with the Weyl f-structure \overline{W}^{f}.

We should remark that this invariant is produced by an affine transformation on A_{3}^{1}(M) of an operator of CInv(3).

The invariant tensor D characterizes the flatness of the Weyl generalized spaces. Theorem 5.2.3 Let (M, ĝ, W^{f})be a Weyl generalized manifold, ∇ semi-symmetric connection, compatible with the Weyl f-structure W^{f}. Then

1. There exist local the 1-forms p and q such that the space endowed with the semi-symmetric connection given by

(5.2.1) \overline{∇}_{fX}Y = ∇_{fX}Y + q(Y)X + p(X)Y − g(X, Y)q^{#} is flat;

2. The tensor D is vanishing are equivalent. Acknowledgement

The results presented here comprise the major part of the author’s doctoral thesis at the University ”AL.I. CUZA”, Iaşi (22 of January 1999), co-directed by Prof.Dr. Constantin Udrişte (University POLITEHNICA of Bucharest) and Acad.Prof.Dr.Doc. Radu Miron (University ”AL.I. CUZA”, Iaşi). I would like to express my gratitude to the advisors who guided this work, for their valuable comments and very useful suggestions.

A version of this paper was presented at the Workshop on Electromagnetic Flows and Dynamics, Oct. 15-19, 1999, University Politehnica of Bucharest, supported by M.E.N. Grant No 21815/28.09.98, C.N.S.U.-31.

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University of Bucharest, Faculty of Mathematics Department of Geometry, 14 Academiei Str., 70109 Bucharest 1, Romania e-mail: ihirica@geometry.math.unibuc.ro