{'fraction_non_alphanumeric': 0.04476995305164319, 'fraction_numerical': 0.017652582159624414, 'mean_word_length': 4.174115818111154, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 2, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 3, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
On geometry of congruences of null strings in 4-dimensional complex and real pseudo-Riemannian spaces

4-dimensional spaces equipped with 2-dimensional (complex holomorphic or real smooth) completely integrable distributions are considered. The integral manifolds of such distributions are totally null and totally geodesics 2-dimensional surfaces which are called the null strings. Properties of congruences (foliations) of such 2-surfaces are studied. Some relations between properties of congruences of null strings, Petrov-Penrose type of SD Weyl spinor and algebraic types of the traceless Ricci tensor are analyzed.


Introduction
The present paper is devoted to some geometrical aspects of complex and real 4dimensional spaces. We assume that spaces are equipped with the holomorphic (in complex case) or smooth (in real case) metric. Except the metric tensor we assume the existence of 2-dimensional, completely integrable distributions. Their integral manifolds are totally null and totally geodesic 2-dimensional surfaces, the null strings. The family of such surfaces constitute the congruence (foliation) of the null strings. 2-dimensional distribution is related to the corresponding 2-form, which is self-dual (SD) or anti-selfdual (ASD). In this paper we analyze the SD congruences of the null strings. In what follows we use abbreviation cns (cns := congruence of the SD null strings).
The idea of m-dimensional real spaces equipped with n-dimensional (n < m) totally null, parallely propagated distributions appeared in the fifties [32] and such spaces are known nowadays as Walker spaces. For our purposes the most interesting is the case with m = 4 and n = 2, but in general we do not assume that the distribution is parallel. In dimension 4, the only real spaces which admit cns are spaces equipped with the metric of the neutral (split, ultrahyperbolic) signature (+ + −−). Moreover, such structures are admitted by 4-dimensional complex manifolds.
In complex geometry cns play a great role. These objects appear in the theory of heavenly spaces (H-spaces) and in the twistor theory as a α(β)-surfaces. The relation between cns and the algebraic degeneration of SD Weyl spinor in Einstein spaces has been found in [19]. The further analysis of the spaces which admit cns led to the concept of the hyperheavenly spaces (HH-spaces). Some properties of cns have been studied in [2,22] but the significant progress in understanding the geometry of cns has been done in [21,27]. In [21] the properties of cns are related to the properties of shear-free null geodesic congruences (abbreviate by sngc). Clearly, cns are complex analogs of sngc. They have been considered as the most important structures in so-called Plebański programme. The main idea of the Plebański programme was to find the general techniques of generating the real solutions of the Einstein field equations from the complex ones. Unfortunately, such techniques are still unknown despite over 40 years of thorough investigations.
Real 4-dimensional Walker spaces equipped with 2-dimensional integrable and totally null distributions have been recently analyzed intensively (see [3,6] and references therein). It appeared, that the most natural formalism in description of the 4-dimensional Walker spaces is the spinorial formalism. Spinorial formalism as a mathematical tool has been developed since the sixties [14][15][16]18]. Spaces with both SD and ASD 2-dimensional integrable distributions have been introduced in [6] (see also [9]). Recently some 4dimensional spaces equipped with the metric of neutral signature and two distinct cns appeared naturally in the works devoted to geometrical model of bodies which roll on each other without slipping and twisting [10,11].
Cns play an important role in Lorentzian geometry as well. The main idea of distinguished papers [13,31] was to treat sngc as an intersection of the SD and ASD congruences of the null strings which exist in the complexification of the tangent bundle. In [31] Robinson manifold have been introduced and their relation with cns has been pointed out. Also the SD null string equations (3.2) have been used to obtain a special class of solutions of the Rarita-Schwinger equation [29].
Cns play also a great role in the generalizations of the Goldberg -Sachs theorem [8]. For complex, 4-dimensional Einstein spaces the analog of Goldberg -Sachs theorem was presented in [19]. Then this idea has been generalized to the case of nonzero traceless Ricci tensor [25] and to the lower [12] and higher dimensions [30]. In [26] the authors considered "sharp" versions of the Goldberg -Sachs theorem in dimension 4.
All mentioned above facts prove that cns carry a very important structure which deserves closer analysis. Our previous work [4] was devoted to Einstein para-Hermite and Einstein para-Kähler spaces. Such spaces are equipped with two distinct cns (see also [1] and references therein). In this paper we analyze the non-Einsteinian case. There are two main goals of the present paper. The first is to analyze the relation between objects which characterize cns (expansion, Sommers vector) and algebraic types of the traceless Ricci tensor (in complex case small portion of such analysis has been presented in [22]). The second aim is to analyze the spaces equipped with three [27] or four distinct cns.
Our considerations are purely local. The results are valid for the 4-dimensional complex manifolds and can be easily carried over to the case of the real manifold with neutral signature metric. Usually it is enough to replace all the holomorphic functions by real smooth functions and complex coordinates by the real ones. Sometimes there appear subtle differences between complex and real spaces and these differences are pointed out and thoroughly analyzed.
The paper is organized, as follows.
Section 2 contains basic information on the spinorial formalism. We use spinorial formalism in Infeld -Van der Waerden -Plebański notation rather then Penrose notation. In section 3 we consider the spaces which admit one cns. Relation between properties of the cns and algebraic types of the traceless Ricci tensor are pointed out. Finally a new Theorem 3.7 concerning the relation between cns and null eigenvectors of the traceless Ricci tensor is proved. Particularly, we find all algebraic types of the traceless Ricci tensor generated by the one nonexpanding cns. Subsection devoted to the Weyl spinor is less original and it contains results known earlier. However, these results are spread all over the wide literature, so we present them here for completeness.
In section 4 we analyze the spaces equipped with two distinct cns. It appears, that two distinct cns determine the traceless Ricci tensor completely (4.7) and it is the main result of this section. All algebraic types of the traceless Ricci tensor generated by two distinct cns are found. Finally, the relation between null eigenvectors of the traceless Ricci tensor and properties of two distinct cns is proved.
Section 5 is devoted to spaces which admit a richer structure then the para-Hermite spaces, i.e. they admit the existence of three or four distinct cns. We find the general form of the metric which admits two expanding cns and one nonexpanding cns (metric (5.13)). Finally, we consider the case which -according to our best knowledge -has not been considered earlier: the space equipped with four distinct cns. In this case we arrive at the metric (5.21) with constraint equations (5.22). Concluding remarks end the paper.

Formalism
In this section we present foundations of the spinorial formalism which seem to be crucial in geometry based on cns. For more detailed treatment see [16,18,20].
Denote by Λ p (M) the space of all p-forms on M (p = 0, 1, 2, 3, 4). Let (e 1 , e 2 , e 3 , e 4 ) ∈ Λ 1 (M) be the members of the null tetrad so the metric can be written in the form The bases of the Λ p (M) are given by (A = 1, 2,Ḃ =1,2) Nevertheless, one has to carefully raise and lower spinorial indices in the objects from tangent space. Indeed, if ∂Ȧ means ∂/∂ΨȦ and ΨȦ =∈ȦḂ ΨḂ then consistency with (2.4) implies The 1-forms g AḂ are holomorphic in CR. In the case of RR the 1-forms g AḂ have the following properties under complex conjugation: where the overbar stands for the complex conjugation.
where ω ∈ Λ(M) := ⊕ 4 p=0 Λ p (M) is the p-form and it has the local representation Operation * defined by (2.7) is the idempotent operation on Λ(M): * * ω = ω. Under (2.7) the p-forms defined by (2.2) behave as follows * 1 = vol , * g AḂ =ǧ AḂ , * S AB = S AB , * SȦḂ = −SȦḂ (2.9) * ǧ AḂ = g AḂ , * vol = 1 Some identities involving (2.2) are helpful in calculations The connection 1-forms Γ ab in null tetrad formalism are connected with the spinorial connection 1-forms Γ AB and ΓȦḂ by the relations 11) The connection 1-forms can be decomposed according to The general formula for the spinorial covariant derivative of the arbitrary spinor field Ψ AḂ CḊ reads and the matrices g aAḂ are defined by the relation g AḂ = g AḂ a e a . Note that we have The arbitrary vector V has the form The first and second Cartan structure equations read and RȦḂ are the components of curvature 2-forms of the connection Γ A B or ΓȦḂ, respectively, D := − 1 2 g AḂ ∇ AḂ and d := − 1 2 g AḂ ∂ AḂ . The 2-forms R AB = R (AB) and RȦḂ = R (ȦḂ) can be decomposed with respect to the bases S AB and SȦḂ as follows The objects C ABCD = C (ABCD) , CȦḂĊḊ = C (ȦḂĊḊ) , C ABĊḊ = C (AB)ĊḊ = C AB(ĊḊ) and R have a transparent geometrical meaning. Namely, C ABCD is the spinorial image of the self-dual (SD) part of the Weyl tensor, CȦḂĊḊ is the spinorial image of the anti-self-dual (ASD) part of the Weyl tensor, C ABĊḊ is the spinorial image of the traceless Ricci tensor and, finally, R is the curvature scalar. Algebraic classification of totally symmetric 4-index spinors (like C ABCD and CȦḂĊḊ) has been presented in [14].  [26] the authors used the following symbols for these types: G r , SG, G, II r , II, D r , D, III r , N r and 0, respectively. We do not present here the details of this classification. They can be found in [5,26].
Note, that under the complex conjugation we have CR : Γ AB , ΓȦḂ , C ABCD , CȦḂĊḊ , C ABĊḊ and R are holomorphic HR : The Ricci identities for 1-index spinors read From (2.19) we can easily obtain the Ricci identities for any spinor Ψ A...Ḃ... C...Ḋ... . Let m A and µ A be the pair of linearly independent spinors such that m A µ A = 1. SD curvature coefficients C (i) , i = 1, ..., 5 are defined as follows The relation between the traceless Ricci tensor C ab and its spinorial image C ABĊḊ reads We decompose the traceless Ricci tensor C ABĊḊ according to where AṀṄ , BṀṄ and CṀṄ are symmetric, 2-index dotted spinors (see [16] for more details). From those spinors the following scalars can be constructed The characteristic polynomial of the matrix (C a b ) of the traceless Ricci tensor reads x 4 − C [1] x 3 + C [2] x 2 − C [3] x + C [4] (2.25) where coefficients C After long but elementary calculations we find the relations between C and the scalars given by (2.23) Definitions of undotted and dotted Plebański spinors are [16,17] Plebański spinors are 4-index and totally symmetric, so they can be classified analogously like SD and ASD Weyl spinors. The algebraic classification of the traceless Ricci tensor plays an important role in analysis presented in this paper. Such classification in 4-dimensional spaces with Lorentzian metric has been done in [16,17] (15 types) and then it was generalized on the complex case in [24] (17 types). There are 33 different types of the traceless Ricci tensor in 4dimensional real spaces equipped with the metric of signature (+ + −−). The detailed discussion which leads to this classification can be found in [5]. Here we present only brief summary of the results of [5]. The information about the algebraic type of the traceless Ricci tensor in UR is gathered in the following symbol Inside the square bracket all different eigenvalues E i , i = 1, 2, ..., N 0 of the polynomial W(x) together with their multiplicities n i are listed. Of course The last equality follows from the fact, that the matrix (C a b ) is traceless. Complex eigenvalues are denoted by Z and the real ones by R. Real eigenvalues have additional superscript which denotes the type of the corresponding eigenvector. R s means, that the eigenvector which corresponds to the eigenvalue R is space-like, R t -time-like, R n -null, R ns -null or space-like, R nt -null or time-like and finally R nst means, that the eigenvector can be of the arbitrary type. [We use the same definition of space-like and time-like vectors, like in the Lorentzian case, i.e. V a V a > 0 means, that V a is space-like, V a V a < 0 stands for time-like vectors and finally, V a V a = 0 means, that the vector is null]. Superscript v denotes the number of eigenvectors and the numbers q i in round bracket describes the form of the minimal polynomial. Namely, the minimal polynomial of the matrix (C a b ) has the form In what follows we often abbreviate the cns defined by the 2-form Σ by Σ-congruence. The 2-form Σ which satisfies the conditions (3.1a) and (3.1b) from the Theorem 3.2 has the form Σ = m A m B S AB where m A is nowhere vanishing spinor. The condition (3.1c) implies, that 2-form Σ is an element of the 2-surface completely integrable in the Frobenius sense. It gives restrictions on the spinor m A . Indeed, one finds that spinor m A has to satisfy the following equations The crucial equations (3.2) are called the SD null string equations. From (3.2) we find where Z AṀ is the Sommers vector [21] and the spinor MṀ is the expansion of the congruence of the SD null strings (see [21]). With fixed Riemannian structure the expansion describes the most important and invariant property of the cns. If MṀ = 0 then the 2-dimensional distribution D m A := {m A aḂ, m A bḂ}, aḂbḂ = 0 is parallely propagated. It means, that ∇ X V ∈ D m A for every vector field V ∈ D m A and for arbitrary vector field X. Such cns are called nonexpanding or plane. If MṀ = 0 then we deal with expanding (or deviating) cns. Under re-scaling m A = mm A the expansion and Sommers vector transform as follows

SD Weyl spinor
Consider the integrability conditions of the SD null strings equations. Acting on (3.3) with ∇ MṄ and using Ricci identities (2.19), one arrives at the equations Immediately we find Theorem 3.3. If a spinor m A generates a congruence of SD null strings, then it is a Penrose spinor.
Proof. Contracting the integrability conditions of the SD null strings equations Define spinor µ A by the relation µ A m A = 1. Then spinors m A and µ A constitute the basis of the 1-index undotted spinors. The SD curvature coefficients C (i) , i = 1, ..., 5 are defined as usual by (2.20). Eqs. (3.5a) constitute the set of 6 equations which relate SD Weyl spinor and the curvature scalar with the Sommers vector and the expansion of the cns. We easily conclude that Note that the coefficient C (5) remains undetermined, so the existence of only one cns does not determine SD conformal curvature. Using (3.6), the integrability conditions (3.5a) reduce to the equations From (3.7) (with help of (3.6)) we find the curvature scalar Finally (3.7) leaves us with one constraint equation Gathering: from 6 equations (3.5a) we get the formulas (3.6) for curvature coefficients C (1) , C (2) , C (3) and C (4) , the curvature scalar R given by (3.8) and one constraint equation (3.9).
Petrov-Penrose types of C ABCD of the spaces equipped with one cns are gathered in the

Traceless Ricci tensor
Now we find the possible types of the traceless Ricci tensor which are admitted by the space equipped with one cns in CR and UR. The criteria described in [24] allow to establish the algebraic type of the traceless Ricci tensor in CR. In [5] we presented the similar criteria for the Plebański-Przanowski types of the traceless Ricci tensor in UR.
If a space is equipped with one cns then the traceless Ricci tensor is determined with precision up to the factor m A C ABṀṄ and it is given by (3.5b). Using the decomposition (2.22) we find BȦḂ and CȦḂ as Spinor AȦḂ is not determined and, consequently, the spaces equipped with one cns only admit all Plebański-Przanowski types of the traceless Ricci tensor. Stronger restrictions on the traceless Ricci tensor appear if spaces admit the nonexpanding cns. Indeed, one finds that MȦ = 0 =⇒ CȦḂ = 0 =⇒ n = c = s = 0 what implies that the only nonzero scalars C [i] are C [2] = 4b , C [4] = 4b 2 (3.11) The characteristic polynomial (2.25) takes the form The Plebański spinors read Using criteria given in [5] and the form of the Plebański spinors we find that UR spaces equipped with one nonexpanding cns admit only 12 types of the traceless Ricci tensor (see Table 2). In the case of the CR there are only 9 different types of the traceless Ricci tensor.
There is an interesting connection between the existence of nonexpanding cns and the eigenvalues of the characteristic polynomial of the traceless Ricci tensor. Indeed, we have Theorem 3.5 (Przanowski, [22]). In CR the existence of nonexpanding congruence of SD null strings implies that the characteristic polynomial of the traceless Ricci tensor has two double or one quadruple eigenvalue, namely λ = ± √ −2b.
Corollary 3.6 (Przanowski, [22]). In CR the existence of nonexpanding congruence of SD null strings implies that the traceless Ricci tensor has at least one null eigenvector which is tangent to the null strings. Moreover: • if there is exactly one null eigenvector then it is tangent to the null string • if there are two linearly independent null eigenvectors then at least one of them is tangent to the null string • if there are three or four linearly independent null eigenvectors then exactly two of them are tangent to the null string Remark. In UR Theorem 3.5 still holds true but Corollary 3.6 does not hold true anymore. Indeed, there are two algebraic types (namely [ (11) ) which are generated by the nonexpanding congruence of SD null strings, but they do not admit real null eigenvector of C a b . However, the following theorem holds true in both CR and UR (slightly weaker version of this theorem has been presented in [22]). Theorem 3.7. Let µ A rḂ be a null eigenvector of the traceless Ricci tensor and let m A be any spinor such that m A µ A = 0. Then m A rḂ is also a null eigenvector of the traceless Ricci tensor iff CȦḂrḂ = 0 where CȦḂ is defined by m A and µ A according to (2.22).
Proof. Without any loss of generality we put µ A m A = 1. Because the vector µ A rȦ is null eigenvector of the traceless Ricci tensor then From (3.14) if follows that Remark. Obviously, Theorem 3.7 holds true for the CȦḂ = 0, i.e., for the nonexpanding cns and in this form it was presented in [22]. However, for the proof it is enough to assume CȦḂrḂ = 0 =⇒ c = 0. It means, that Theorem 3.7 holds true for special class of expanding cns such that c = 0.
In Table 2 we gather information about the number of null eigenvectors of the traceless Ricci tensor in UR and CR. 2 T T means, that both null eigenvectors are tangent to the null string and 2 T N means, that only one of them is tangent to the null string. In CR there is subtle difference between types (2) [4N ] a 2 and (2) [4N ] b 2 and this difference has not been recognized earlier in [22].
Conversely, consider two spinors m A and µ A which generate cns such that m A µ A = 0 . Then m A µ A = 0 =⇒ Σ ∧ Σ = 0.
Spaces which admit two distinct congruences of the SD (or ASD) null strings are known as a (complex or real) para-Hermite spaces [7,23]. Spinors m A and µ A can be normalized m A µ A = 1 without any loss of generality. We deal now with the equations for the Σ-congruence and Σ-congruence and with the integrability conditions of the equations (4.1a) and (4.1b)

we find the relation between the expansions and Sommers vectors for the Σ-and Σ-congruences
In what follows we use the Eq. (4.3) to eliminate Z AḂ . Integrability conditions (4.2a)-(4.2b) constitute the set of 12 equations. One quickly finds that all curvature coefficients C (i) are determined by Eqs. (4.2a) and (4.2b) reduces to the following ones Eqs. (4.5a) and (4.5b) have to be consistent with each other and with (4.4). After simple calculations we find that the curvature scalar R has the form (3.8). We are left with three constraint equations which can be rearranged into the form Integrability conditions (4.2c)-(4.2d) give 12 equations for the 9 components of the traceless Ricci tensor. Three of these equations are identically satisfied and the rest bring us to the formulas  (2.20) and remembering that C (1) = C (5) = 0 we find αC (i) = 0 = βC (i) , i = 2, 3, 4. All C (i) cannot be simultaneously equal zero, so the only solution is α = β = 0 what contradicts the assumption that spinor α A in nonzero.
Because C (1) = C (5) = 0 one arrives to the following formula for SD Weyl spinor All possible types and forms of C ABCD are gathered in the Table 3 in which the following abbreviation has been used Petrov-Penrose types of C ABCD in para-Hermite spaces depends strongly on expansions of the congruences (compare formulas (4.5)). For example, if both congruences are nonexpanding, then C (2) = 0 = C (4) and 6C (3) = R. Consequently, 2C ABCD = R m (A m B µ C µ D) so for R = 0 we have type [D] in CR or type [D r ] in UR and for R = 0 we have type [−]. In the case of Einstein spaces, the conditions C ab = 0 and R = −4Λ reduce the possible Petrov-Penrose types of C ABCD . In this case analysis is less straightforward (it is necessary to use the Bianchi identities) and we do not present the details here. Some portion of this analysis can be found in [23]. The results are gather in Tables 4 and 5.

Traceless Ricci tensor
The traceless Ricci tensor in para-Hermite spaces is completely determined via Sommers vector, expansions of both congruences and covariant derivatives of these objects (compare Eqs. (4.7)). The most general case is when both congruences are expanding. It seems (somehow surprisingly) that in this case all Plebański-Przanowski types of the traceless Ricci tensor are admitted. The case with one nonexpanding congruence has been discussed in subsection 3.3 and this discussion remains valid in spaces equipped with one Types Conditions SD Weyl spinor CR UR  However, these are not the only restrictions on the traceless Ricci tensor implied by the existence of two nonexpanding cns. We have transparent corollary which follows from the Theorem 3.7 Corollary 4.4. Assume, that the traceless Ricci tensor generated by two distinct, nonexpanding congruences of SD null strings admits a null eigenvector. Then • there are two linearly independent null eigenvectors, the first is tangent to the Σcongruence, the second to the Σ-congruence • there are four linearly independent null eigenvectors, the first pair is tangent to the Σ-congruence, the second pair to the Σ-congruence Corollary 4.4 proves, that the cases with one null eigenvector or three linearly independent null eigenvectors of the traceless Ricci tensor are not admitted in the spaces equipped with two nonexpanding cns.
Using algebraic criteria given [5], the form of Plebański spinors and Corollary 4.4 we arrive at the Table 6.  5 Spaces equipped with three or four congruences of SD null strings

General analysis
In this section we investigate the spaces which admit even richer structure then the para-Hermite spaces, i.e. we consider the spaces admitting three or four distinct congruences of SD null strings. As a starting point we take (complex or real) para-Hermite space equipped with two distinct congruences Σ and Σ (generated by the spinors m A and µ A , respectively, such that µ A m A = 1). Consider now another cns, say Σ-congruence, Σ = s A s B S AB . Let Σ-congruence be complementary to the congruences Σ and Σ: Σ ∧ Σ = 0, Σ ∧ Σ = 0. Decomposing the spinor s A according to the formula [4N ] 1 Table 6: Possible types of the traceless Ricci tensor in para-Hermite spaces equipped with two nonexpanding congruence of the SD null strings.
we find that m = 0 and µ = 0, otherwise the complementary conditions are not satisfied. As a consequence of existence of the Σ-congruence we find Acting on (5.2) with s M ∇ MḂ we find the integrability conditions of the equations (5.2)  Table 3 Table 3). However, if δ = 0 (in CR) or δ > 0 (in UR) the equation (5.3) has two different nonzero solutions. One finds, that  Table 3).
Summing up: the ratio ξ can be found as a solution of the quadratic equation (5.3) and it depends on the curvature coefficients of the SD Weyl spinor C (2) , C (3) and C (4) .
where ξ is given by (5.4

5.2
The metric of the spaces equipped with three distinct congruences of SD null strings The general metric of the space equipped with three distinct cns has been found in [27]. The authors considered a para-Hermite space equipped with the congruences Σ  The special case with one of these congruences being nonexpanding has not been considered in [27]. In this case we arrive at the following theorem: Theorem 5.3. If a complex (real) space admits three distinct congruences of SD null strings and one of them is nonexpanding, then the metric of this space can be brought to the form where Ω = Ω(zȦ), fṀ = fṀ (z A , zḂ) and (z A , zḂ) are local coordinates.
Proof. The space admits three distinct cns, so the metric has the form (5.11). Detailed calculations prove, that the expansion of the Σ-congruence is proportional to the

5.3
The metric of the spaces equipped with four distinct congruences of SD null strings  Then we arrive at the following possible types of tensor of matter admitted by such spaces are unknown. The problem of Lorentzian manifolds with 4 distinct sngc is very interesting and we are going to analyze it in details in near future.
Quite surprisingly, the space with two distinct expanding cns seems to admit all algebraic types of the traceless Ricci tensor. We believe, that in fact there are some restrictions which reduce the number of possible types in such a case. Similarly, in spaces equipped with three or four expanding cns those restrictions should be even stronger. Frankly, this problem is more advanced and it involves deeper analysis.
Finally, expansion of the cns is the most transparent property of such structure and has clear geometrical interpretation. It seems, that the Sommers vector carries some information about cns. Natural question arises if the properties the Sommers vector define a new subclassification of cns?