Family of Projective Projections on Tensors and Connections

C.Udrişte and I.E. Hirică

Abstract

§1 finds the explicit expressions for all projective projections on the set of (1,2)-tensors. §2 analyses the action of extended projective projections on the set of connections and shows that in particular one gets the classical Thomas connection. §3 gives properties of the almost projective transformations of connections.

Mathematics Subject Classification: 15A72, 53A55, 53B10

Key words: projective projections, tensor space decomposition, connection space decomposition

Introduction

The theory of invariant decompositions of tensors and connections using global projections built with the Kronecker δ-tensor or the δ-tensor together with the Riemannian metric, almost complex structure, almost contact structure etc have been initiated by the first author in 1975. It was discussed by letters (1976-1977) with Prof.Dr. Lieven Vanhecke and was orally communicated as remarks at different Conferences in Geometry. A part of this theory is detailed in this paper on (1,2)-tensors and connections, but of course it can be generalized for (p, q)-tensors. The most interesting generalization is to apply the theory for the curvature (1,3)-tensor, and to relate the projection on connections with the projection on the corresponding curvature tensors, but this subject will be developed in another paper.

As was remarked by Krupka [4], [6] whose invariant trace decompositions are special cases of ours, the results can be applied in the representation theory of the orthogonal group, developed by Weyl [12]. Extensive literature on this subject can be found from different perspectives. For examples, N.Bokan [1] considers the case of a torsion free connection on a space endowed with a positive definite metric and finds a decomposition of the underlying tensor space, invariant with respect to the group SO(n).

Balkan Journal of Geometry and Its Applications, Vol.2, No.2, 1997, pp. 139-156 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

1 Family of projective projections on (1,2)-tensors

Let V be a real n-dimensional vector space, where n ≥ 2, T_{2}^{1}(V) = {T_{bc}^{a}} be the vector space of all tensors T of type (1,2), δ_{j}^{i} be the symbol of Kronecker, I = {δ_{a}^{r}δ_{s}^{b}δ_{t}^{c}} be the identity on T_{2}^{1}(V).

A projection P = {P_{a}^{bc} _{st}^{r}} on T_{2}^{1}(V) of the form

P_{a}^{bc} _{st}^{r} = x_{1}δ_{a}^{r}δ_{s}^{b}δ_{t}^{c} + x_{2}δ_{a}^{b}δ_{s}^{r}δ_{t}^{c} + x_{3}δ_{a}^{r}δ_{s}^{c}δ_{t}^{b} + x_{4}δ_{a}^{c}δ_{s}^{b}δ_{t}^{r} + x_{5}δ_{a}^{c}δ_{s}^{r}δ_{t}^{b} + x_{6}δ_{a}^{b}δ_{s}^{c}δ_{t}^{r}

is called a projective projection. The adjective ”projective” is justified by the fact that there exist induced projections P which transform a symmetric connection into the Thomas projective connection (see Section 2).

Of course, P is a projection iff P^{2} = P or P_{a}^{bc} _{st}^{r}P_{r}^{st} _{jk}^{i} = P_{a}^{bc} _{jk}^{i}, i.e.,

(1.1)  \{ \begin{matrix}x_{1}^{2} + x_{3}^{2} = x_{1} \\ 2x_{1}x_{2} + nx_{2}^{2} + x_{2}x_{5} + x_{3}x_{5} + x_{2}x_{6} + x_{3}x_{6} + nx_{5}x_{6} = x_{2} \\ 2x_{1}x_{3} = x_{3} \\ 2x_{1}x_{4} + x_{3}x_{6} + nx_{4}^{2} + x_{4}x_{6} + x_{3}x_{5} + x_{4}x_{5} + nx_{5}x_{6} = x_{4} \\ 2x_{1}x_{5} + x_{2}x_{3} + x_{2}x_{4} + x_{3}x_{4} + nx_{4}x_{5} + nx_{2}x_{5} + x_{5}^{2} = x_{5} \\ 2x_{1}x_{6} + x_{2}x_{3} + x_{2}x_{4} + nx_{2}x_{6} + x_{3}x_{4} + nx_{4}x_{6} + x_{6}^{2} = x_{6}. \end{matrix}

This algebraic system is easily obtained via the simplified expression P = x_{1}I_{1} + ... + x_{6}I_{6}, the condition P^{2} = P and the table of compositions

\begin{matrix}* & I_{1} & I_{2} & I_{3} & I_{4} & I_{5} & I_{6} \\ I_{1} & I_{1} & I_{2} & I_{3} & I_{4} & I_{5} & I_{6} \\ I_{2} & I_{2} & nI_{2} & I_{6} & I_{6} & I_{2} & nI_{6} \\ I_{3} & I_{3} & I_{5} & I_{1} & I_{6} & I_{2} & I_{4} \\ I_{4} & I_{4} & I_{5} & I_{5} & nI_{4} & nI_{5} & I_{4} \\ I_{5} & I_{5} & nI_{5} & I_{4} & I_{4} & I_{5} & nI_{4} \\ I_{6} & I_{6} & I_{2} & I_{2} & nI_{6} & nI_{2} & I_{6} \end{matrix}

If P = {P_{a}^{bc} _{st}^{r}} is a projective projection, then its supplement Q = I − P is also a projective projection. That in way the following theorem is true. Theorem 1.1.. If (x_{1}^{0}, ..., x_{6}^{0}) is a solution for the algebraic system (1.1), then (1 − x_{1}^{0}, −x_{2}^{0}, ..., −x_{6}^{0}) is also a solution.

The projective projection P belongs to the class of invariant tensors studied by D.Krupka and J.Janyska [5] (a tensor T ∈ T_{r}^{r}(V) being invariant iff A ∘ T = T, for any A ∈ GL(V)).

Let us solve the system (1.1). For that reason we start with

(1.2)  \{ \begin{matrix}x_{1}^{2} + x_{3}^{2} = x_{1} \\ 2x_{1}x_{3} = x_{3}. \end{matrix}

This system is equivalent to

(1.3) \{ \begin{matrix}x_{1} + x_{3} = 0 \\ 2x_{1}x_{3} = x_{3} \end{matrix} or (1.4) \{ \begin{matrix}x_{1} + x_{3} = 1 \\ 2x_{1}x_{3} = x_{3}. \end{matrix}

From (1.3) we obtain \{ \begin{matrix}x_{1} = 0 \\ x_{3} = 0 \end{matrix} or \{ \begin{matrix}x_{1} = \frac{1}{2} \\ x_{3} = −\frac{1}{2}. \end{matrix} . The supplementary solutions

\{ \begin{matrix}x_{1} = 1 \\ x_{3} = 0 \end{matrix} or \{ \begin{matrix}x_{1} = \frac{1}{2} \\ x_{3} = \frac{1}{2}. \end{matrix} are obtained from (1.4). We solve the initial system (1.1)

for x_{1} = 0, x_{3} = 0 and for x_{1} = x_{3} = \frac{1}{2}. The other two cases are obtained taking supplementary solutions for the system (1.1).

I. In the case x_{1} = x_{3} = 0, from (1.1) we get the system

(1.5)  \{ \begin{matrix}x_{2}(1 − nx_{2} − x_{5}) = x_{6}(x_{2} + nx_{5}) \\ x_{5}(1 − nx_{2} − x_{5}) = x_{4}(x_{2} + nx_{5}) \\ x_{4}(1 − nx_{4} − x_{6}) = x_{5}(x_{4} + nx_{6}) \\ x_{6}(1 − nx_{4} − x_{6}) = x_{2}(x_{4} + nx_{6}). \end{matrix}

Multiplying the first equation with the third equation and the second equation with the fourth equation, we get

(1.6)  \{ \begin{matrix}x_{2}x_{4}(1 − nx_{2} − x_{5})(1 − nx_{4} − x_{6}) = x_{5}x_{6}(x_{2} + nx_{6})(x_{4} + nx_{6}) \\ x_{5}x_{6}(1 − nx_{2} − x_{5})(1 − nx_{4} − x_{6}) = x_{2}x_{4}(x_{2} + nx_{5})(x_{4} + nx_{6}). \end{matrix}

A). We study the case x_{2}x_{4} − x_{5}x_{6} ≠ 0. From (1.5) we obtain

\{ \begin{matrix}nx_{2} + x_{5} = 1 \\ x_{2} + nx_{5} = 0 \end{matrix} and \{ \begin{matrix}x_{4} + nx_{6} = 0 \\ x_{6} + nx_{4} = 1. \end{matrix}

We get the solution

\{ \begin{matrix}x_{2} = x_{4} = −\frac{n}{1 − n^{2}} \\ x_{5} = x_{6} = \frac{1}{1 − n^{2}} . \end{matrix}

B). We study the case x_{2}x_{4} = x_{5}x_{6}. B1). Let x_{2}x_{4} = x_{5}x_{6} ≠ 0. The system (1.6) implies

(1 − nx_{2} − x_{5})(1 − nx_{4} − x_{6}) = (x_{2} + nx_{5})(x_{4} + nx_{6}).

We find the system

\{ \begin{matrix}n(x_{2} + x_{4}) + x_{5} + x_{6} = 1 \\ x_{2}x_{4} = x_{5}x_{6}. \end{matrix}

B2). Let x_{2}x_{4} = x_{5}x_{6} = 0.

a) If x_{2} = x_{5} = 0, then (1.5) becomes \{ \begin{matrix}x_{4}(1 − nx_{4} − x_{6}) = 0 \\ x_{6}(1 − nx_{4} − x_{6}) = 0. \end{matrix} We get x_{2} =

x_{4} = x_{5} = x_{6} = 0 or \{ \begin{matrix}x_{2} = x_{5} = 0 \\ 1 = nx_{4} + x_{6}. \end{matrix}

142 C.Udrişte and I.E.Hirică

b) If x_{2} = x_{6} = 0, then (1.5) implies \{ \begin{matrix}x_{5}(1 − x_{5} − nx_{4}) = 0 \\ x_{4}(1 − x_{5} − nx_{4}) = 0. \end{matrix} We find x_{2} = x_{4} =

x_{5} = x_{6} = 0 or \{ \begin{matrix}x_{2} = x_{6} = 0 \\ 1 = nx_{4} + x_{5}. \end{matrix}

c) If x_{4} = x_{5} = 0, then (1.5) becomes \{ \begin{matrix}x_{2}(1 − nx_{2} − x_{6}) = 0 \\ x_{6}(1 − nx_{2} − x_{6}) = 0. \end{matrix} Consequently

x_{2} = x_{4} = x_{5} = x_{6} = 0 or \{ \begin{matrix}x_{4} = x_{5} = 0 \\ 1 = nx_{2} + x_{6}. \end{matrix}

d) If x_{4} = x_{6} = 0, then (1.5) gives \{ \begin{matrix}x_{2}(1 − nx_{2} − x_{5}) = 0 \\ x_{5}(1 − nx_{2} − x_{5}) = 0. \end{matrix} We get x_{2} = x_{4} =

x_{5} = x_{6} = 0 or \{ \begin{matrix}x_{4} = x_{6} = 0 \\ 1 = nx_{2} + x_{5}. \end{matrix}

II. The case x_{1} = x_{3} = \frac{1}{2}. From (1.1) we get

(1.7)  \{ \begin{matrix}nx_{2}^{2} + x_{2}x_{5} + \frac{1}{2}x_{5} + \frac{1}{2}x_{6} + x_{2}x_{6} + nx_{5}x_{6} = 0 \\ nx_{4}^{2} + x_{4}x_{6} + \frac{1}{2}x_{5} + \frac{1}{2}x_{6} + x_{4}x_{6} + nx_{5}x_{6} = 0 \\ \frac{1}{2}x_{2} + \frac{1}{2}x_{4} + x_{2}x_{4} + nx_{4}x_{5} + nx_{2}x_{5} + x_{5}^{2} = 0 \\ \frac{1}{2}x_{2} + \frac{1}{2}x_{4} + x_{2}x_{4} + nx_{2}x_{6} + nx_{4}x_{6} + x_{6}^{2} = 0. \end{matrix}

From the first two equations of the system (1.7) we obtain

(x_{2} − x_{4})[(x_{2} + x_{4})n + x_{5} + x_{6}] = 0.

From the last equations of the system (1.7) we obtain (x_{5}−x_{6})[(x_{2}+x_{4})n+x_{5}+x_{6}] = 0.

Let

\{ \begin{matrix}x_{2} = x_{4} \\ x_{5} = x_{6}. \end{matrix}

From the first equation and the third equation of (1.7)

\{ \begin{matrix}nx_{2}^{2} + 2x_{2}x_{5} + x_{5} + nx_{5}^{2} = 0 \\ x_{2}^{2} + 2nx_{2}x_{5} + x_{2} + x_{5}^{2} = 0 \end{matrix}

we get (x_{2} − x_{5})[(n − 1)(x_{2} − x_{5}) − 1] = 0. If x_{2} = x_{5}, then x_{2}[(2n + 2)x_{2} + 1] = 0. We obtain

(1.8_{1}) \{ \begin{matrix}x_{2} = x_{4} = 0 \\ x_{5} = x_{6} = 0, \end{matrix}

or

(1.8_{2})  x_{2} = x_{4} = x_{5} = x_{6} = −\frac{1}{2(n + 1)}.

If x_{2} = x_{5} + \frac{1}{n − 1}, then we obtain the solution

\{ \begin{matrix}x_{2} = x_{4} = \frac{1}{2(n − 1)} \\ x_{5} = x_{6} = \frac{1}{2(1 − n)} \end{matrix} or \{ \begin{matrix}x_{2} = x_{4} = \frac{1}{n^{2} − 1} \\ x_{5} = x_{6} = \frac{n}{1 − n^{2}} . \end{matrix}

The system

\{ \begin{matrix}n(x_{2} + x_{4}) + x_{5} + x_{6} = 0 \\ nx_{2}^{2} + x_{2}x_{5} + \frac{1}{2}x_{6} + \frac{1}{2}x_{5} + x_{2}x_{6} + nx_{5}x_{6} = 0 \\ \frac{1}{2}x_{2} + \frac{1}{2}x_{4} + x_{2}x_{4} + nx_{4}x_{5} + nx_{2}x_{5} + x_{5}^{2} = 0 \end{matrix}

is equivalent to

\{ \begin{matrix}n(x_{2} + x_{4}) + x_{5} + x_{6} = 0 \\ x_{5}x_{6} = \frac{1}{2}(x_{2} + x_{4}) + x_{2}x_{4}. \end{matrix}

Theorem 1.2. The solutions of the quadratic system (1.1) and hence the set of all projective projections P on T_{2}^{1}(V) are given by

I.

a) x_{1} = x_{2} = x_{3} = x_{4} = x_{5} = x_{6} = 0.

b) x_{1} = 0, x_{3} = 0, x_{2} = x_{4} = \frac{n}{n^{2} − 1}, x_{5} = x_{6} = \frac{1}{1 − n^{2}} ;

c) x_{1} = x_{3} = 0, x_{2}x_{4} = x_{5}x_{6}, 1 = n(x_{2} + x_{4}) + x_{5} + x_{6}.

Introducing the parameters x_{5} = λ, x_{6} = μ and imposing the condition (\frac{λ + μ − 1}{n})^{2} ≥

4λμ, λ, μ ∈ R, the values x_{2} = α, x_{4} = β are solutions of the equation z^{2} + \frac{1}{n}(λ + μ − 1)z + λμ = 0.

The supplementary solutions for I are I′.

a) x_{1} = 1, x_{3} = x_{2} = x_{4} = x_{5} = x_{6} = 0;

b) x_{1} = 1, x_{3} = 0, x_{2} = x_{4} = \frac{n}{1 − n^{2}} , x_{5} = x_{6} = \frac{1}{n^{2} − 1};

c) x_{1} = 1, x_{3} = 0, x_{2}x_{4} = x_{5}x_{6}, −1 = n(x_{2} + x_{4}) + x_{5} + x_{6}.

Denoting x_{5} = λ, x_{6} = μ, and imposing the condition (\frac{λ + μ + 1}{n})^{2} ≥

4λμ, λ, μ ∈ R, the values x_{2} = α, x_{4} = β are solutions of the equation z^{2} + \frac{1}{n}(λ + μ + 1)z + λμ = 0.

II.

a) x_{1} = x_{3} = \frac{1}{2}, x_{2} = x_{4} = x_{5} = x_{6} = 0;

b) x_{1} = x_{3} = \frac{1}{2}, x_{2} = x_{4} = x_{5} = x_{6} = −\frac{1}{2(n + 1)};

c) x_{1} = x_{3} = \frac{1}{2}, x_{2} = x_{4} = −x_{5} = −x_{6} = \frac{1}{2(n − 1)};

d) x_{1} = x_{3} = \frac{1}{2}, x_{2} = x_{4} = \frac{1}{n^{2} − 1}, x_{5} = x_{6} = \frac{n}{1 − n^{2}} .

e) x_{1} = x_{3} = \frac{1}{2}, n(x_{2} + x_{4}) + x_{5} + x_{6} = 0, x_{5}x_{6} = \frac{1}{2}(x_{2} + x_{4}) + x_{2}x_{4}.

With x_{5} = λ, x_{6} = μ, [\frac{1}{n}(λ + μ) − 1]^{2} ≥ 1 + 4λμ, λ, μ ∈ R, the values x_{2} =

α, x_{4} = β are solutions of the equation z^{2} + \frac{1}{n}(λ + μ)z + λμ + \frac{1}{2n}(λ + μ) = 0. The supplementary solutions for II are II′.

a) x_{1} = −x_{3} = \frac{1}{2}, x_{2} = x_{4} = x_{5} = x_{6} = 0;

b) x_{1} = −x_{3} = \frac{1}{2}, x_{2} = x_{4} = x_{5} = x_{6} = \frac{1}{2(n + 1)};

c) x_{1} = −x_{3} = \frac{1}{2}, x_{2} = x_{4} = −x_{5} = −x_{6} = \frac{1}{2(1 − n)};

d) x_{1} = −x_{3} = \frac{1}{2}, x_{2} = x_{4} = \frac{1}{1 − n^{2}} , x_{5} = x_{6} = \frac{n}{n^{2} − 1}.

e) x_{1} = −x_{3} = \frac{1}{2}, n(x_{2} + x_{4}) + x_{5} + x_{6} = 0, x_{5}x_{6} = −\frac{1}{2}(x_{2} + x_{4}) + x_{2}x_{4}.

With x_{5} = λ, x_{6} = μ, [\frac{1}{n}(λ + μ) + 1]^{2} ≥ 1 + 4λμ, λ, μ ∈ R, the values x_{2} =

α, x_{4} = β are solutions of the equation z^{2} + \frac{1}{n}(λ + μ)z + λμ − \frac{1}{2n}(λ + μ) = 0.

The images of a (1,2)-tensor by the precedent projections are obvious and contain the following generalizations of the results of Krupka [4]. Theorem 1.3. Let T = (T_{bc}^{a}) ∈ T_{2}^{1}(V). There exists an infinity of projective projections P = (P_{a}^{bc} _{st}^{r}) such that Ω = P T = (P_{a}^{bc} _{st}^{r}T_{bc}^{a}) is a traceless tensor (i.e. Ω_{st}^{s} = Ω_{ts}^{s} = 0).

I. T_{at}^{a} = 0.

1) Ω = 0 for x_{1} = x_{3} = x_{4} = x_{5} = 0, nx_{2} + x_{6} = 1;

2) Ω_{st}^{r} = T_{st}^{r} − \frac{1}{1 − n^{2}} (T_{ta}^{a}δ_{s}^{r} − nT_{sa}^{a}δ_{t}^{r}) for

x_{1} = 1, x_{3} = 0, x_{6} + nx_{2} = 0, x_{4} = \frac{n}{1 − n^{2}} , x_{5} = \frac{1}{n^{2} − 1};

3) Ω_{st}^{r} = \frac{1}{2}(T_{st}^{r} +T_{ts}^{r})+δ_{s}^{r}x_{5}T_{ta}^{a} −δ_{t}^{r}T_{sa}^{a}(\frac{1}{2} +nx_{5}), for x_{1} = x_{3} = \frac{1}{2}, \frac{1}{2} +x_{4} +nx_{5} =

0, n(x_{2} + x_{4}) + x_{5} + x_{6} = 0, x_{5}x_{6} = \frac{1}{2}(x_{2} + x_{4}) + x_{2}x_{4}, \frac{1}{2} + x_{6} + nx_{2} ≠ 0;

4) Ω_{st}^{r} = \frac{1}{2}(T_{st}^{r}−T_{ts}^{r})+x_{5}δ_{s}^{r}T_{ta}^{a}+(\frac{1}{2}−nx_{5})δ_{t}^{r}T_{sa}^{a}, for x_{1} = −x_{3} = \frac{1}{2}; −\frac{1}{2}+x_{4}+nx_{5} =

0, n(x_{2} + x_{4}) + x_{5} + x_{6} = 0, x_{5}x_{6} = −\frac{1}{2}(x_{2} + x_{4}) + x_{2}x_{4}, \frac{1}{2} + x_{6} + nx_{2} ≠ 0.

II. T_{at}^{a} = T_{ta}^{a} = 0.

1) Ω = T, for x_{1} = 1, x_{2} = x_{3} = x_{4} = x_{5} = x_{6} = 0;

2) Ω = 0, for x_{1} = x_{3} = 0, x_{2} = x_{4} = \frac{n}{n^{2} − 1}, x_{5} = x_{6} = \frac{1}{1 − n^{2}} ;

Family of Projective Projections on Tensors and Connections 145

3) Ω_{st}^{r} = \frac{1}{2}(T_{st}^{r} + T_{ts}^{r}), for x_{1} = x_{3} = \frac{1}{2}, x_{2} = x_{4} = −x_{5} = −x_{6} = \frac{1}{2(n − 1)};

4) Ω_{st}^{r} = \frac{1}{2}(T_{st}^{r} − T_{ts}^{r}), for x_{1} = −x_{3} = \frac{1}{2}, x_{2} = x_{4} = x_{5} = x_{6} = \frac{1}{2(n + 1)}.

III. T_{at}^{a} = −T_{ta}^{a}.

1) Ω_{st}^{r} = \frac{1}{2}(T_{st}^{r} + T_{ts}^{r}), for x_{1} = x_{3} = \frac{1}{2}, x_{2} = x_{4} = x_{5} = x_{6} = 0;

2) Ω_{st}^{r} = \frac{1}{2}(T_{st}^{r} − T_{ts}^{r}) + \frac{1}{1 − n}(δ_{s}^{r}T_{at}^{a} − δ_{t}^{r}T_{as}^{a}), for x_{1} = −x_{3} = \frac{1}{2}, x_{5} = x_{6} =

\frac{n}{n^{2} − 1}, x_{2} = x_{4} = \frac{1}{1 − n^{2}} .

IV. T_{at}^{a} = T_{ta}^{a}.

1) Ω_{st}^{r} = \frac{1}{2}(T_{st}^{r} + T_{ts}^{r}) − \frac{1}{n + 1}(δ_{s}^{r}T_{at}^{a} + δ_{t}^{r}T_{as}^{a}), for x_{1} = x_{3} = \frac{1}{2}, x_{2} = x_{4} =

\frac{1}{n^{2} − 1}, x_{5} = x_{6} = \frac{n}{1 − n^{2}} ;

2) Ω_{st}^{r} = \frac{1}{2}(T_{st}^{r} − T_{ts}^{r}), for

a) x_{1} = −x_{3} = \frac{1}{2}, x_{2} = x_{4} = x_{5} = x_{6} = 0;

b) x_{1} = −x_{3} = \frac{1}{2}, x_{2} = x_{4} = −x_{5} = −x_{6} = \frac{1}{2(1 − n)};

V. T_{ta}^{a} = −\frac{x_{6} + nx_{2}}{x_{4} + nx_{5}} T_{at}^{a} ⇒ Ω = 0, for

x_{4} ≠ −nx_{5} x_{2}x_{4} = x_{5}x_{6}, 1 = n(x_{2} + x_{4}) + x_{5} + x_{6}.

VI.

T_{ta}^{a} = −\frac{1 + x_{6} + nx_{2}}{x_{4} + nx_{5}} T_{at}^{a} ⇒ Ω_{st}^{r} = T_{st}^{r} − \frac{x_{4}}{x_{4} + nx_{5}} T_{as}^{a}δ_{t}^{r} − \frac{x_{5}}{x_{4} + nx_{5}} T_{at}^{a}δ_{s}^{r},

for

x_{4} ≠ −nx_{5}, x_{2}x_{4} = x_{5}x_{6}, −1 = n(x_{2} + x_{4}) + x_{5} + x_{6}.

VII.

T_{ta}^{a} = −\frac{1 + 2x_{6} + 2nx_{2}}{1 + 2x_{4} + 2nx_{5}} T_{at}^{a}.

Ω_{st}^{r} = \frac{1}{2}(T_{st}^{r} + T_{ts}^{r}) + \frac{x_{6} + x_{2}}{1 + 2x_{4} + 2nx_{5}} T_{as}^{a}δ_{t}^{r} − \frac{x_{4} + x_{5}}{1 + 2x_{4} + 2nx_{5}} T_{at}^{a}δ_{s}^{r},

for

x_{1} = x_{3} = \frac{1}{2}, n(x_{2} + x_{4}) + x_{5} + x_{6} = 0, x_{5}x_{6} = \frac{1}{2}(x_{2} + x_{4}) + x_{2}x_{4}, x_{4} + nx_{5} ≠ −\frac{1}{2}.

VIII.

T_{ta}^{a} = −\frac{1 + 2x_{6} + 2nx_{2}}{1 + 2x_{4} + 2nx_{5}} T_{at}^{a}.

Ω_{st}^{r} = \frac{1}{2}(T_{st}^{r} − T_{ts}^{r}) + \frac{x_{4} − x_{5}}{−1 + 2x_{4} + 2nx_{5}} (T_{at}^{a}δ_{s}^{r} − T_{as}^{a}δ_{t}^{r}),

for

x_{1} = \frac{1}{2}, x_{3} = −\frac{1}{2}, n(x_{2}+x_{4})+x_{5}+x_{6} = 0, x_{5}x_{6} = −\frac{1}{2}(x_{2}+x_{4})+x_{2}x_{4}, x_{4}+nx_{5} ≠ \frac{1}{2}.

IX. For any T having arbitrary traces one gets

1) Ω_{st}^{r} = T_{st}^{r} + \frac{1}{n^{2} − 1} [δ_{s}^{r}(−nT_{at}^{a} + T_{ta}^{a}) + δ_{t}^{r}(T_{as}^{a} − nT_{sa}^{a}], for

x_{1} = 1, x_{3} = 0, x_{2} = x_{4} = \frac{n}{1 − n^{2}} , x_{5} = x_{6} = \frac{1}{n^{2} − 1}.

2) Ω_{st}^{r} = δ_{s}^{r}(x_{2}T_{at}^{a} + \frac{1}{1 − n^{2}} T_{ta}^{a}) + δ_{t}^{r}(−nx_{2}T_{as}^{a} + \frac{n}{1 − n^{2}} T_{sa}^{a}), for

x_{1} = x_{3} = 0, x_{5} = \frac{1}{1 − n^{2}} , x_{4} = \frac{n}{n^{2} − 1}, x_{6} = −nx_{2};

3) Ω = 0, for x_{1} = x_{2} = x_{3} = x_{4} = x_{5} = x_{6} = 0;

4) Ω_{st}^{r} = T_{st}^{r}+δ_{s}^{r}x_{2}T_{at}^{a}−δ_{t}^{r}(1+nx_{2})T_{as}^{a}, for x_{1} = 1, x_{3} = x_{4} = x_{5} = 0, nx_{2}+x_{6}+1 = 0;

5) Ω_{st}^{r} = \frac{1}{2}(T_{st}^{r} + T_{ts}^{r}) − \frac{1}{2(n + 1)}(T_{at}^{a} + T_{ta}^{a})(δ_{s}^{r} + δ_{t}^{r}), for

x_{1} = x_{3} = \frac{1}{2}, x_{2} = x_{4} = x_{5} = x_{6} = −\frac{1}{2(n + 1)};

6) Ω_{st}^{r} = \frac{1}{2}(T_{st}^{r} +T_{ts}^{r})+δ_{s}^{r}[x_{2}T_{at}^{a} + \frac{1}{2(1 − n)}T_{ta}^{a}]+δ_{t}^{r}[−(\frac{1}{2} +nx_{2})T_{as}^{a} + \frac{1}{2(n − 1)}T_{sa}^{a}],

for

x_{1} = x_{3} = \frac{1}{2}, nx_{2} + x_{6} + \frac{1}{2} = 0, x_{4} = −x_{5} = \frac{1}{2(n − 1)};

7) Ω_{st}^{r} = \frac{1}{2}(T_{st}^{r} +T_{ts}^{r})+δ_{s}^{r}[x_{2}T_{at}^{a} + \frac{1}{2(1 + n)}T_{ta}^{a}]+δ_{t}^{r}[−(\frac{1}{2} +nx_{2})T_{as}^{a} + \frac{1}{2(n + 1)}T_{sa}^{a}],

for

x_{1} = −x_{3} = \frac{1}{2}, x_{6} = −nx_{2} − \frac{1}{2}, x_{4} = x_{5} = \frac{1}{2(n + 1)};

8) Ω_{st}^{r} = T_{st}^{r} + δ_{s}^{r}(\frac{n}{1 − n^{2}} T_{at}^{a} − nx_{4}T_{ta}^{a}) + +δ_{t}^{r}(\frac{1}{n^{2} − 1}T_{as}^{a} + x_{4}T_{ta}^{a}), for

x_{1} = 1, x_{3} = 0, x_{2} = \frac{n}{1 − n^{2}} , x_{6} = \frac{1}{n^{2} − 1}, x_{5} = −nx_{4}.

9) Ω_{st}^{r} = T_{st}^{r} + δ_{s}^{r}(1 − nx_{4})T_{at}^{a} + +δ_{t}^{r}x_{4}T_{ta}^{a}), for

x_{1} = x_{3} = 0, x_{2} = x_{6} = 0, x_{5} = 1 − nx_{4}.

Proof. Ω is a traceless tensor iff

(1.9)  \{ \begin{matrix}(x_{1} + x_{6} + nx_{2})T_{at}^{a} + (x_{3} + x_{4} + nx_{5})T_{ta}^{a} = 0 \\ (x_{1} + nx_{4} + x_{5})T_{ta}^{a} + (x_{3} + nx_{6} + x_{2})T_{at}^{a} = 0. \end{matrix}

This system with the unknowns T_{at}^{a}, T_{ta}^{a} is compatible. We study all the cases of the theorem 1.2.

I′ a) The system (1.9) is equivalent to T_{at}^{a} = T_{ta}^{a} = 0. We get II, 1);

b) We obtain the case IX, 1);

c) 1) x_{4} = −nx_{5}. 1.1. 1 + x_{6} + nx_{2} = 0. In this case we find IX, 4) and 8); 1.2. T_{at}^{a} = 0 and 1 + x_{6} + nx_{2} ≠ 0. We get I, 2).

2) If x_{4} ≠ −nx_{5} we obtain VI. I. a) We get IX, 3);

b) The system (1.9) is equivalent to T_{ta}^{a} = T_{at}^{a} = 0. We get II, 2);

c) 1) If x_{4} + nx_{5} ≠ 0, then we obtain V.;

2) x_{4} + nx_{5} = 0. If T_{at}^{a} = 0 and x_{6} + nx_{2} ≠ 0, then we get I, 1). If x_{6} + nx_{2} = 0, then we arrive at IX, 2) and 9).

II. a) The system becomes T_{ta}^{a} + T_{at}^{a} = 0. We obtain III, 1);

b) (1.9) is identic satisfied and we have IX, 5);

c) (1.9) is equivalent with T_{ta}^{a} = T_{at}^{a} = 0. We get II, 3);

d) (1.9) becomes T_{ta}^{a} = T_{at}^{a} and we obtain IV, 1)

e) 1) \frac{1}{2} + x_{4} + nx_{5} = 0. We get I, 3) for T_{at}^{a} = 0 and IX, 6) for \frac{1}{2} + x_{6} + nx_{2} = 0.

2) If \frac{1}{2} + x_{4} + nx_{5} ≠ 0, then we find VII. II′. a) The system (1.9) is equivalent to T_{ta}^{a} = T_{at}^{a}. We get the case IV, 2), a);

b) (1.9) becomes T_{ta}^{a} = T_{at}^{a} = 0. We have the case II, 4);

c) (1.9) is equivalent with T_{ta}^{a} = T_{at}^{a}. We get the case IV, 2), b);

d) (1.9) becomes T_{ta}^{a} = −T_{at}^{a}. We get the case III, 2);

e) 1) x_{4} + nx_{5} = \frac{1}{2};

1.1. If T_{at}^{a} = 0, we get I, 4);

1.2. If x_{6} + nx_{2} = −\frac{1}{2}, then we get the case IX, 7);

2) If x_{4} + nx_{5} ≠ −\frac{1}{2}, we get VIII. Remark 1.1. a) If T is a traceless tensor, then Ω = P T is a traceless tensor, for any projective projection P.

b) The theorem ” Let V be a real n-dimensional vector space, where n ≥ 2 and let A = (A_{kl}^{i}) ∈ T_{2}^{1}(V). Then there exist a unique traceless tensor B = (B_{kl}^{i}) ∈ T_{2}^{1}(V) and unique 1-forms C = (C_{k}), D = (D_{k}) ∈ Λ^{1}(V), such that A_{kl}^{i} = B_{kl}^{i}+δ_{l}^{i}D_{k}+δ_{k}^{i}C_{l}, where

C_{l} = \frac{1}{n^{2} − 1}(nA_{tl}^{t} − A_{lt}^{t}, D_{k} = \frac{1}{n^{2} − 1}(−A_{tk}^{t} + nA_{kt}^{t}),

B_{kl}^{i} = A_{kl}^{i} − \frac{1}{n^{2} − 1}[δ_{k}^{i}(nA_{tl}^{t} − A_{lt}^{t}) + δ_{l}^{i}(−A_{tk}^{t} + nA_{kt}^{t})], ”

proved by Krupka in [4], is a particular case of ours (IX, 1). Its trace decomposition problem corresponds to the case x_{1} = 1, x_{3} = 0 for our projective projections.

2 Family of projective projections on affine connections

Let M be a differentiable n-dimensional manifold and T_{2}^{1}M be the bundle of (1,2)- tensor fields over M. The previous projection P extends to a global projection field on T_{2}^{1}M denoted also by P whose extended coefficients x_{1}, ..., x_{6} are scalar fields. Some of the scalar fields x_{1}, .., x_{6} are arbitrary functions, others depend on the these arbitrary functions, and some of them are constant functions.

Denote with A_{2}^{1}(M) the set of all geometrical objects of type (1, 2) whose difference is a (1, 2)-tensor field. The set A_{2}^{1}(M) is an affine vector space modelled on the vector space T_{2}^{1}(M). Obviously, the set C of all affine connections on M and T_{2}^{1}(M) are affine subspaces of A_{2}^{1}(M). Any projection on T_{2}^{1}(M) induces a projection on A_{2}^{1}(M).

Let Γ = {Γ_{jk}^{i}} be an affine connection on M. The projective projections P of Theorem 1.2 work on C by the rule Π = P Γ = (P_{a}^{bc} _{st}^{r}Γ_{bc}^{a}). They produce almost projective connections Π iff x_{1} + x_{3} = 1. Otherwise (x_{1} + x_{3} = 0), the image P (C) consists of geometrical objects fields Π of type (1,2) which are not connections; particularly the

torsion tensor \frac{1}{2}(Γ_{jk}^{i} − Γ_{kj}^{i}) is the image of Γ by the projective projection P having the coefficients

x_{1} = \frac{1}{2}, x_{3} = −\frac{1}{2}, x_{2} = x_{4} = x_{5} = x_{6} = 0.

Theorem 2.1. Let x_{1} = 1, x_{3} = 0. The images of the projective projections P on C consist of the almost projective connections

Π_{st}^{r} = Γ_{st}^{r} + δ_{s}^{r}ψ_{t} + δ_{t}^{r}φ_{s},

where φ_{s} and ψ_{t} are defined by a) For x_{2} = x_{4} = x_{5} = x_{6} = 0, ψ_{t} = φ_{t} = 0;

b) For x_{2} = −x_{4} \frac{n}{1 − n^{2}} , x_{5} = x_{6} = \frac{1}{n^{2} − 1},

ψ_{t} = \frac{1}{n^{2} − 1}(−nΓ_{at}^{a} + Γ_{ta}^{a}), φ_{s} = \frac{1}{n^{2} − 1}(Γ_{as}^{a} − nΓ_{sa}^{a});

c) x_{5} = λ, x_{6} = μ, (\frac{λ + μ + 1}{n})^{2} ≥ 4λμ, λ, μ ∈ R, x_{2} = α, x_{4} = β solutions of

the equation z^{2} + \frac{1}{n}(λ + μ + 1)z + λμ = 0,

ψ_{t} = αΓ_{at}^{a} + λΓ_{ta}^{a}, φ_{s} = βΓ_{sa}^{a} + μΓ_{as}^{a}.

Remark. The geometrical objects φ_{s} and ψ_{t} are not 1-forms. Corollary 2.1. Let x_{1} = 1, x_{3} = 0 and P the corresponding projective projections acting on symmetric affine connections. The images of P consist of the almost projective connections

Π_{st}^{r} = Γ_{st}^{r} + δ_{s}^{r}ψ_{t} + δ_{t}^{r}φ_{s},

where ψ_{t} and φ_{s} are related to Γ_{t} = Γ_{at}^{a} in the following ways:

Case b) ⇒ ψ_{t} = φ_{t} = −\frac{1}{n + 1}Γ_{t}.

Case a) ⇒ ψ_{s} = φ_{s} = 0.

Case c) ⇒ ψ_{t} = (α + λ)Γ_{t}, φ_{s} = (β + μ)Γ_{s}.

In particular, for the case c) with α + λ = β + μ = −\frac{1}{n + 1} and for the case b) we

find the Thomas projective connection [11]

Π_{st}^{r} = Γ_{st}^{r} − \frac{1}{n + 1}(δ_{s}^{r}Γ_{t} + δ_{t}^{r}Γ_{s}),

corresponding to the projection

P_{a}^{bc} _{st}^{r} = δ_{a}^{r}δ_{s}^{b}δ_{t}^{c} − \frac{1}{n + 1}δ_{a}^{b}δ_{s}^{r}δ_{t}^{c} − \frac{1}{n + 1}δ_{a}^{b}δ_{s}^{c}δ_{t}^{r}.

Theorem 2.1′. Let x_{1} = x_{3} = 0. The images of the projective projections P on C consist of the objects of type (1,2)

Π_{st}^{r} = δ_{s}^{r}ψ_{t} + δ_{t}^{r}φ_{s},

where φ_{s} and ψ_{t} are defined by a) for x_{2} = x_{5} = x_{4} = x_{6} = 0,

ψ_{t} = φ_{t} = 0

b) for x_{2} = x_{4} = \frac{n}{n^{2} − 1}, x_{5} = x_{6} = \frac{1}{1 − n^{2}} ,

ψ_{t} = \frac{1}{n^{2} − 1}(nΓ_{at}^{a} − Γ_{ta}^{a}), φ_{s} = \frac{1}{n^{2} − 1}(nΓ_{sa}^{a} − Γ_{as}^{a});

c) for x_{5} = −λ, x_{6} = −μ, (\frac{1 − λ − μ}{n})^{2} ≥ 4λμ, λ, μ ∈ R, x_{2} = −α, x_{4} = −β

solutions of the equation z^{2} − \frac{1}{n}(1 − λ − μ)z + λμ = 0,

ψ_{t} = −αΓ_{at}^{a} − λΓ_{ta}^{a}, ψ_{s} = −βΓ_{sa}^{a} − μΓ_{as}^{a}.

Theorem 2.2. Let x_{1} = x_{3} = \frac{1}{2}. The images of the projective projections P on C consist of the almost projective connections

Π_{st}^{r} = \frac{1}{2}(Γ_{st}^{r} + Γ_{ts}^{r}) + δ_{s}^{r}ψ_{t} + δ_{t}^{r}φ_{s},

where φ_{s} and ψ_{t} are defined by a)for x_{2} = x_{4} = x_{5} = x_{6} = 0, ψ_{t} = φ_{t} = 0;

b) for x_{2} = x_{4} = x_{5} = x_{6} = −\frac{1}{2(n + 1)}, ψ_{t} = φ_{t} = −\frac{1}{2(n + 1)}(Γ_{at}^{a} + Γ_{ta}^{a});

c) for x_{2} = x_{4} = −x_{5} = −x_{6} = \frac{1}{2(n − 1)},

ψ_{t} = −φ_{t} = \frac{1}{2(n − 1)}(Γ_{at}^{a} − Γ_{ta}^{a});

d) for x_{2} = x_{4} = \frac{1}{n^{2} − 1}, x_{5} = x_{6} = \frac{n}{1 − n^{2}} ,

ψ_{t} = \frac{1}{n^{2} − 1}(Γ_{at}^{a} − nΓ_{ta}^{a}), φ_{s} = \frac{1}{n^{2} − 1}(Γ_{sa}^{a} − nΓ_{as}^{a});

e) for x_{5} = λ, x_{6} = μ, [\frac{1}{n}(λ + μ) − 1]^{2} ≥ 1 + 4λμ, λ, μ ∈ R, x_{2} = α, x_{4} = β

solutions of the equation z^{2} + \frac{1}{n}(λ + μ)z + λμ + \frac{1}{2n}(λ + μ) = 0,

ψ_{t} = αΓ_{at}^{a} + λΓ_{ta}^{a}, φ_{s} = βΓ_{sa}^{a} + μΓ_{as}^{a}.

Corollary 2.2. Let x_{1} = x_{3} = \frac{1}{2} and P the corresponding projective projections working on symmetric connections. The images of P consist of the next almost projective connections

a) x_{2} = x_{4} = x_{5} = x_{6} = 0 ⇒ Π_{st}^{r} = Γ_{st}^{r}

b) x_{2} = x_{4} = x_{5} = x_{6} = −\frac{1}{2(n + 1)} ⇒ Π_{st}^{r} = Γ_{st}^{r} − \frac{1}{n + 1}(δ_{s}^{r}Γ_{t} + δ_{t}^{r}Γ_{s})

c) x_{2} = x_{4} = −x_{5} = −x_{6} = \frac{1}{2(n − 1)} ⇒ Π_{st}^{r} = Γ_{st}^{r};

d) x_{2} = x_{4} = \frac{1}{n^{2} − 1}, x_{5} = x_{6} = \frac{n}{1 − n^{2}} ⇒ Π_{st}^{r} = Γ_{st}^{r} − \frac{1}{n + 1}(δ_{s}^{r}Γ_{t} + δ_{t}^{r}Γ_{s}).

e) x_{5} = λ, x_{6} = μ, x_{2} = α, x_{4} = β, α, β, λ, μ ∈ R, satisfying

n(α + β) + λ + μ = 0, λμ = −\frac{1}{2}(α + β) + αβ ⇒

⇒ Π_{st}^{r} = Γ_{st}^{r} + δ_{s}^{r}(λ + α)Γ_{t} + δ_{t}^{r}(β + μ)Γ_{s}).

Remark 2.1. In particular, the case e) λ + α = β + μ = −\frac{1}{n + 1} and the cases b)

and d) produce the Thomas projective connection.

Theorem 2.2′. Let x_{1} = \frac{1}{2}, x_{3} = −\frac{1}{2}. The images of the projective projections P on C consist of the objects of type (1,2)

Π_{st}^{r} = \frac{1}{2}(Γ_{st}^{r} − Γ_{ts}^{r}) + δ_{s}^{r}ψ_{t} + δ_{t}^{r}φ_{s},

where φ_{s} and ψ_{t} are defined by:

a) for x_{2} = x_{4} = 0, x_{5} = x_{6} = 0, ψ_{t} = φ_{t} = 0;

b) for x_{2} = x_{4} = x_{5} = x_{6} = \frac{1}{2(n + 1)}, φ_{t} = ψ_{t} = \frac{1}{2(n + 1)}(Γ_{at}^{a} + Γ_{ta}^{a});

c) for x_{2} = x_{4} = −x_{5} = −x_{6} = \frac{1}{2(1 − n)},

ψ_{t} = −φ_{t} = \frac{1}{2(1 − n)}(Γ_{at}^{a} − Γ_{ta}^{a});

d) for x_{2} = x_{4} = \frac{1}{1 − n^{2}} , x_{5} = x_{6} = \frac{n}{n^{2} − 1},

ψ_{t} = \frac{1}{1 − n^{2}} (Γ_{at}^{a} − nΓ_{ta}^{a}), φ_{s} = \frac{1}{1 − n^{2}} (Γ_{sa}^{a} − nΓ_{as}^{a});

e) for x_{5} = −λ, x_{6} = −μ, [\frac{1}{n}(λ + μ) + 1]^{2} ≥ 1 + 4λμ, λ, μ ∈ R, x_{2} = −λ, x_{4} =

−β solutions of the equations z^{2} + \frac{1}{n}(λ + μ)z − \frac{1}{2n}(λ + μ) + λμ = 0,

ψ_{t} = −αΓ_{at}^{a} − λΓ_{ta}^{a}, φ_{s} = −βΓ_{sa}^{a} − μΓ_{as}^{a}.

3 Almost projective transformations of connections

Let M be a finite dimensional differentiable manifold endowed with the affine connection Γ. The class Γ̄ of the almost projective transformations (apt) of the connection Γ is defined by [3], [9]

Γ̄ = Γ + η ⊗ I + I ⊗ ξ,

where η, ξ ∈ ∧^{1}(M). Theorem 3.1. For each connection Γ and each projective projection P in Theorems 2.1, 2.2, 2.1′, 2.2′ there exists a class of connections Γ̄ satisfying the commutative diagram

Γ \underrightarrow{apt} Γ̄
P ↓  ↓ P
Π = Π̄,

where Π = P Γ, Π̄ = P Γ̄. This diagram reflects also the invariance of Π with respect to Γ̄ (the gauge invariance of Π with respect to the projective group). Proof. We fix a projective projection P = (P_{a}^{bc} _{st}^{r}) by (x_{1}, ..., x_{6}). Since

Γ̄_{bc}^{a} = Γ_{bc}^{a} + η_{b}δ_{c}^{a} + ξ_{c}δ_{b}^{a}

it is enough to prove that there exist the 1-forms η = (η_{b}), ξ = (ξ_{c}) such that

P_{a}^{bc} _{st}^{r}(η_{b}δ_{c}^{a} + ξ_{c}δ_{b}^{a}) = [(x_{1} + nx_{4} + x_{6})η_{s} + (x_{3} + x_{4} + nx_{6})ξ_{s}]δ_{t}^{r}+
+[(x_{2} + x_{3} + nx_{5})η_{t} + (x_{1} + nx_{2} + x_{5})ξ_{t}]δ_{s}^{r} = 0.

This condition is equivalent to the linear system

\{ \begin{matrix}(x_{1} + nx_{4} + x_{6})η_{s} + (x_{3} + x_{4} + nx_{6})ξ_{s} = 0 \\ (x_{2} + x_{3} + nx_{5})η_{s} + (x_{1} + nx_{2} + x_{5})ξ_{s} = 0, \end{matrix}

with 2n unknows (η_{1}, ..., η_{n}, ξ_{1}, ..., ξ_{n}) and with 2n equations. The determinant of the matrix of this linear system is

Δ = −[(x_{1} + nx_{4} + x_{6})(x_{1} + nx_{2} + x_{5}) − (x_{2} + x_{3} + nx_{5})(x_{3} + x_{4} + nx_{6})]^{n}.

For each (x_{1}, ..., x_{6}) in Theorems 2.1, 2.2, 2.1′, 2.2′ one proves that Δ is as a rule zero, excepting few cases in which Δ ≠ 0. In other words the preceding linear system is as a rule compatible undetermined, excepting few cases in which η = 0, ξ = 0.

In the sequel we suppose that Γ is a symmetric connection, and we identify the connection Γ with the induced covariant derivative ∇. The class of the almost projective transformations of ∇ is characterized by

¯∇_{X}Y = ∇_{X}Y + η(X)Y + ξ(Y)X, ∀X, Y ∈ X (M), η, ξ ∈ ∧^{1}(M).

The curvature tensor fields R̄ of ¯∇ and R of ∇ are related by

R̄(X, Y)Z = R(X, Y)Z − C(Y, Z)X + C(X, Z)Y + dη(X, Y)Z,

where

C(X, Y) = (∇_{X}ξ)(Y) − ξ(X)ξ(Y).

Let (M, g) be a Riemannian space. Denote by g_{ij} the components of the metric g, and by R_{ijkl} the components of the curvature tensor field. Introduce the symbols R · R and Q(g, R) by

(3.1)  \begin{matrix}(R · R)_{hijklm} = −R_{hlm}^{s}R_{sijk} − R_{ilm}^{s}R_{hsjk} − R_{jlm}^{s}R_{hisk} − R_{klm}^{s}R_{hijs}, \\ Q(g, R)_{hijklm} = −g_{mh}R_{lijk} + g_{hl}R_{mijk} − g_{mi}R_{hljk} + g_{il}R_{hmjk}− \\ −g_{jm}R_{hilk} + g_{jl}R_{himk} − g_{km}R_{hijl} + g_{kl}R_{hijm}. \end{matrix}

Pseudo-symmetric manifolds [2], i.e., Riemannian spaces (M, g) for which the fields (∗) R · R, Q(g, R) are linearly dependent at every point of the manifold, constitute a generalization of spaces of constant sectional curvature, along the line of locally symmetric and semi-symmetric spaces R · R = 0, studied by Szabo in [8]), consecutively.

The linear dependence of the fields (∗) is equivalent to

(∗∗)  R · R = LQ(g, R) on U = {x ∈ M|R ≠ R(1) at x},

where

R(1)_{hijk} = \frac{k}{n(n − 1)}(−g_{ik}g_{jh} + g_{ij}g_{kh}),

k being the scalar curvature. Similarly to (3.1) we can define Q(g, A), R · A, Q(D, A), where D, A are tensors of type (0,2).

Let us consider the square matrix whose entries are R_{ijkl}, where ij indicate the rows and kl indicate the colums. The rank of this symmetric matrix will be denoted

by q(x). Obviously q(x) ≤ \frac{n(n − 1)}{2}, ∀x ∈ M ([10]). Theorem 3.2. Let ∇ be the Levi-Civita connection of the Riemannian space (M, g) and ¯∇ its almost projective transformation

¯∇_{X}Y = ∇_{X}Y + η(X)Y + ξ(Y)X,

such that η is a closed 1-form and C = fg, f ∈ F(M). If ¯∇ is a metrique connection (i.e. there exists ḡ ∈ T_{2}^{0}(M), symmetric and positive definite such that ¯∇ḡ = 0) and (M, g) is a pseudo-symmetric manifold with L a constant function, then

(3.2)  (f + L) [ḡ − \frac{1}{n}Trace(ḡ)g] = 0

holds on the open set U. Proof. Because ¯∇ḡ = 0, we get

ḡ_{ij,k} = 2η_{k}ḡ_{ij} + ξ_{i}ḡ_{kj} + ξ_{j}ḡ_{ik},

where the comma denotes covariant differentiation with respect to the Levi-Civita connection. The second covariant derivative is

(3.3)  \begin{matrix}ḡ_{ij,kl} = 2η_{k,l} + ξ_{i,l}ḡ_{kj} + ξ_{j,l}ḡ_{ik} + 2η_{k}(2ξ_{l}ḡ_{ij} + ξ_{i}ḡ_{lj} + ξ_{j}ḡ_{il})+ \\ +ξ_{i}(2η_{l}ḡ_{kj} + +ξ_{k}ḡ_{lj} + ξ_{j}ḡ_{kl}) + ξ_{j}(2η_{l}ḡ_{ik} + ξ_{i}ḡ_{lk} + ξ_{k}ḡ_{il}). \end{matrix}

From (3.3) we get

ḡ_{ij,kl} − ḡ_{ij,lk} = ḡ_{kj}(ξ_{i,l} − ξ_{i}ξ_{l}) + ḡ_{ik}(ξ_{j,l} − ξ_{j}ξ_{l}) − ḡ_{lj}(ξ_{i,k} − ξ_{k}ξ_{i}) − ḡ_{il}(ξ_{j,k} − ξ_{j}ξ_{k}),

which is equivalent to (R · ḡ)_{ijkl} = −Q(C, ḡ)_{ijkl} = −Q(g, fḡ)_{ijkl}.

Using the Theorem 1 of [2] we find

(f + L) [ḡ − \frac{1}{n}Trace(ḡ)g] = 0 on U.

Proposition 3.1. In the same hypothesis of the Theorem 3.2, if moreover (U, g) is not conformally related to (U, ḡ), then (R̄ · R̄)_{ijklm}^{h} = 0 holds on U. Proof.

(R̄ · R̄)_{ijklm}^{h} = R̄_{ijk}^{r}R̄_{rlm}^{h} − R̄_{rkl}^{h}R̄_{ilm}^{r} − R̄_{irk}^{h}R̄_{jlm}^{r} − R̄_{ijr}^{h}R̄_{klm}^{r}.

This relation is equivalent to

(R̄ · R̄)_{ijklm}^{h} = (L + f)Q(g, R)_{ijklm}^{h}.

Using (3.2) we find L = −f and hence (R̄ · R̄)_{ijklm}^{h} = 0 on the set U. Proposition 3.2. In the same hypothesis of the previous Proposition, if moreover ¯∇

is a symmetric connection and the rank of the matrix (R̄_{ijkl}) is q(x) = \frac{n(n − 1)}{2}, then (U, ḡ) has constant curvature. Proof. The proposition is a direct consequence of the Proposition 3.1 and Theorem 2 of [10]. Remark 3.1. If η = ξ, then ¯∇ is the Levi-Civita connection associated to ḡ and hence (M, g) and (M, ḡ) are special geodesically related spaces. The Theorem 3.2 generalizes the Theorem 2 of [2]. In this special case (U, g) has also constant curvature. Theorem 3.3. Let ¯∇ be the almost projective transformation of the Levi-Civita connection of the Riemannian space (M, g),

¯∇_{X}Y = ∇_{X}Y + η(X)Y + ξ(Y)X

such that η is a closed 1-form and ξ a closed, non vanishing 1-form. If (M, g) is an Einstein space and ¯∇-recurrent (i.e. ¯∇_{X} R̄ = ω(X)R̄, ω being a 1-form, and R̄ the curvature tensor field), then the two connections are flat projective (i.e. the projective

curvature tensor W_{jkl}^{i} = R_{jkl}^{i} + \frac{1}{n − 1}(δ_{k}^{i}S_{jl} − δ_{l}^{i}S_{jk}) is 0 and also W̄_{jkl}^{i} = 0, where S

is the Ricci tensor field). Proof. The projective curvature tensor field is invariant with respect to this special almost projective transformation of connections. The relations R̄_{jkl;r}^{i} = ω_{r}R̄_{jkl}^{i} and S̄_{jl;r} = ω_{r}S̄_{jl} imply

(3.4)  W_{jkl;r}^{i} = ω_{r}W_{jkl}^{i}.

The relation (3.4) is equivalent to

(3.5)  W_{ijkl,r} + g_{ir}ξ_{s}W_{jkl}^{s} − ξ_{k}W_{ijrl} − ξ_{l}W_{ijkr} − ξ_{j}W_{irkl} = (ω_{r} + 2η_{r})W_{ijkl}.

Because (M, g) is an Einstein space, we have W_{ijkl} + W_{jikl} = 0. From (3.5) we get

(3.6)  g_{ir}ξ_{s}W_{jkl}^{s} + g_{jr}ξ_{s}W_{ikl}^{s} − ξ_{j}W_{irkl} − ξ_{i}W_{jrkl} = 0

Contracting with g^{ir} in (3.6) we obtain

(3.7)  ξ_{s}W_{jkl}^{s} = 0.

From (3.6) and (3.7) we get ξ_{j}W_{irkl} + ξ_{i}W_{jrkl} = 0 and hence W = W̄ = 0. Theorem 3.3. Let ¯∇ be the almost projective transformation of the affine connection ∇,

¯∇_{X}Y = ∇_{X}Y + kξ(X)Y + ξ(Y)X, k ∈ Z \ \{−1, \frac{n}{2}\} ,

ξ being a closed 1-form. If ∇ and ¯∇ are projective recurrent so that ∇_{X}W = μ(X)W, ¯∇_{X} W̄ = μ(X)W̄, μ being a 1-form, then the two connections ∇, ¯∇ are flat projective. Proof. From the relation

(3.8) \begin{matrix}( ¯∇_{U} W̄)(X, Y)Z = (∇_{U}W)(X, Y)Z + ξ(W (X, Y)Z)U − ξ(X)W (U, Y)Z− \\ −ξ(Y)W (X, U)Z − ξ(Z)W (X, Y)U − 2kξ(U)W (X, Y)Z \end{matrix}

we obtain

(3.9)  2kξ(U)W (X, Y)Z + ξ(X)W (U, X)Z + ξ(Y)W (X, U)Z+
+ξ(Z)W (X, Y)U = ξ(W (X, Y), Z)U.

If {λ^{i}} ⊂ ∧^{1}(M) and {X_{i}} ⊆ X (M) are dual local bases, let us take U = X_{i} in (3.9). Contracting the resulting formula with λ^{i} we get

(3.10)  (n − 2k)ξ(W (X, Y)Z) = 0.

From (3.9) and (3.10) it follows

(3.11) 2kξ(U)W (X, Y)Z + ξ(X)W (U, X)Z + ξ(Y)W (X, U)Z + ξ(Z)W (X, Y)U = 0.

Taking U = Z = X in (3.11) we have

(3.12)  2(k + 1)ξ(X)W (X, Y)X = 0.

There is T ∈ X (M) so that ξ(T) ≠ 0 and hence W (T, Y)T = 0. Using (3.11) we get W (X, Y)Z = 0. Remark 3.2. If we suppose that M is endowed with two affine connections ∇, ¯∇ and A = ¯∇ − ∇, we can construct the deformation algebra U(M, ¯∇, ∇) considering X ⋆ Y = A(X, Y). An element X ∈ U(M, ¯∇, ∇) is called a characteristic vector field if there exists λ ∈ F(M) such that A(X, X) = λX and is called an almost principal vector field if there are f ∈ F(M) and ω ∈ ∧^{1}(M) such that A(Z, X) = fZ + ω(Z)X, ∀X, Z ∈ X (M) [7]. Theorem 3.4. Let ¯∇ be the almost projective transformation of the affine connection ∇, ¯∇ = ∇ + I ⊗ ξ + η ⊗ I, η, ξ being arbitrary 1-forms. All the elements of the deformation algebra U(M, ∇, ¯∇,) are characteristic vector fields and almost principal vector fields.

Acknowledgements. A version of this paper was presented at the First Conference of Balkan Society of Geometers, Politehnica University of Bucharest, September 23-27, 1996.

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University Politehnica of Bucharest Department of Mathematics I Splaiul Independentei 313 77206 Bucharest, Romania e-mail:udriste@mathem.pub.ro

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