arXiv subjects

Paweł Nurowski

Publications and source records attributed to Paweł Nurowski.

10 recordsLinked to original sources

arXiv2026

The Reye geometry inside the 64 lines of the Schur quartic

We identify the classical geometry hidden in the Naskręcki--Pokora $(24_4,32_3)$ configuration on the Schur quartic. Using Höhn's identification of the $24$ selected lines with the $24$ roots of $D_4$, the antipodal involution on the roots induces a fixed-point-free quotient of the incidence configuration, and this quotient is precisely the classical Reye configuration. We also determine the symmetry of the complete $64$-line incidence geometry: its automorphism group has order $4608$, the two Naskręcki--Pokora configurations form a single orbit, and the stabilizer of either has order $2304$ (projectively, $576$). The $64$ lines extend canonically to a $176$-line arrangement carried by six projectively equivalent Schur quartics, with $176=16+16+9\cdot16$ and induced surface permutation group $S_3\times S_3$. Finally, the antipodal quotient itself extends coherently through this six-quartic geometry: on each Schur quartic it produces two Reye configurations sharing the same $16$-element incidence skeleton, and on the full $176$-line arrangement it gives a compatible global quotient. This reveals a precise incidence-theoretic connection with classical desmic geometry, while showing that this connection is not a literal identification with the two Reye configurations arising from the classical desmic construction in $P^3$.

math.AG↗

arXiv2026

What if we decompose a simple tone? The Chinese remainder theorem and structured Levi graphs in music

While motivated by structural problems in mathematical music theory, this article introduces a novel combinatorial framework that advances the classification of cyclic cubic bipartite graphs. We extend the classical study of Levi graphs by endowing their vertices with an internal algebraic anatomy -- specifically, treating them not as empty geometric nodes, but as defined subsets of a cyclic base space Z_n. This internal structure allows us to formalize and classify a highly restricted class of graph isomorphisms: those strictly induced by global affine bijections f(x) = ax+b (mod n) operating directly on the underlying base set. By applying this framework to generalized tone networks (Tonnetze) unrolled via the Chinese Remainder Theorem in composite dimensions -- specifically the classic 12-TET (3x4) and the decaphonic 10-TET (2x5) -- we reveal absolute geometric anchors for these spaces, namely the (9,4) and (6,5) systems respectively. We completely classify the topological orbits of these structured graphs, proving a fundamental architectural dichotomy: while the isomorphic landscape of 12-TET splits into an orientation-preserving family and an orientation-reversing chiral mirror (providing a rigorous foundation for musical Negative Harmony), the 10-TET space is unconditionally orientation-preserving. Finally, we demonstrate that these abstract combinatorial properties manifest as rigidly coherent, parallel auditory universes through explicit structural voice-leading maps and acoustic physical modeling synthesis.

math.CO↗

arXiv2023

Accidental CR structures

We noticed a discrepancy between Élie Cartan and Sigurdur Helgason about the lowest possible dimension in which the simple exceptional Lie group ${\bf E}_8$ can be realized. This raised the question about the lowest dimensions in which various real forms of the exceptional groups ${\bf E}_\ell$ can be realized. Cartan claims that ${\bf E}_6$ can be realized in dimension 16. However Cartan refers to the complex group ${\bf E}_6$, or its split real form $E_I$. His claim is also valid in the case of the real form denoted by $E_{IV}$. We find however that the real forms $E_{II}$ and $E_{III}$ of ${\bf E}_6$ can not be realized in dimension 16 à la Cartan. In this paper we realize them in dimension 24 as groups of CR automorphisms of certain CR structures of higher codimension. As a byproduct of these two realizations, we provide a full list of CR structures $(M,H,J)$ and their CR embeddings in an appropriate ${\bf C}^N$, which satisfy the following conditions: (1) they have real codimension $k>1$, (2) the real vector distribution $H$ proper for the action of the complex structure $J$ is such that $[H,H]+H=TM$, (3) the local group $G_J$ of CR automorphisms of the structure $(M,H,J)$ is simple, acts transitively on $M$ and has isotropy $P$ being a parabolic subgroup in $G_J$, (4) the local symmetry group $G$ of the vector distribution $H$ on $M$ coincides with the group $G_J$ of CR automorphisms of $(M,H,J)$. Because all the CR structures from our list satisfy the last property we call them accidental. Our CR structures of higher codimension with the exceptional symmetries $E_{II}$ and $E_{III}$ are particular entries in this list.

math.CV↗

arXiv2023

Infinitesimal CR Symmetries of Accidental CR Structures

In this companion paper to our article {\em Accidental CR structures} (arxiv.org, January 2023), thought of as an appendix not submitted for publication, we provide complete explicit lists of infinitesimal CR automorphisms for the concerned CR models having respective Lie algebra structures: $${\bf E}_{II}, \qquad\ {\bf E}_{III}, \qquad\ \mathfrak{so}(\ell-1,\ell+1), \qquad\ \mathfrak{su}(p,q).$$ We start from our lists of {\em quadric} CR submanifolds $M^{2n+c} \subset \mathbb{C}^{n+c}$ of codimension $c >1$ which are shown to be {\em accidental}, in the sense that their CR symmetry groups are {\em equal to} (and not smaller than) the symmetry groups of the underlying real distribution structures -- after forgetting the complex structure. Thanks to intensive symbolic computer explorations, we then determine embedded vector field generators of these CR symmetries Lie algebras, and we express them in {\em extrinsic} holomorphic coordinates, because intrinsic formulas would be too extended to be shown.

math.CV↗

arXiv2022

Para-Kähler-Einstein 4-manifolds and non-integrable twistor distributions

We study the local geometry of 4-manifolds equipped with a \emph{para-Kähler-Einstein} (pKE) metric, a special type of split-signature pseudo-Riemannian metric, and their associated \emph{twistor distribution}, a rank 2 distribution on the 5-dimensional total space of the circle bundle of self-dual null 2-planes. For pKE metrics with nonvanishing Einstein constant this twistor distribution has exactly two integral leaves and is `maximally non-integrable' on their complement, a so-called (2,3,5)-distribution. Our main result establishes a simple correspondence between the anti-self-dual Weyl tensor of a pKE metric with non-vanishing Einstein constant and the Cartan quartic of the associated twistor distribution. This will be followed by a discussion of this correspondence for general split-signature metrics which is shown to be much more involved. We use Cartan's method of equivalence to produce a large number of explicit examples of pKE metrics with nonvanishing Einstein constant whose anti-self-dual Weyl tensor have special real Petrov type. In the case of real Petrov type $D,$ we obtain a complete local classification. Combined with the main result, this produces twistor distributions whose Cartan quartic has the same algebraic type as the Petrov type of the constructed pKE metrics. In a similar manner, one can obtain twistor distributions with Cartan quartic of arbitrary algebraic type. As a byproduct of our pKE examples we naturally obtain para-Sasaki-Einstein metrics in five dimensions. Furthermore, we study various Cartan geometries naturally associated to certain classes of pKE 4-dimensional metrics. We observe that in some geometrically distinguished cases the corresponding \emph{Cartan connections} satisfy the Yang-Mills equations. We then provide explicit examples of such Yang-Mills Cartan connections.

math.DG↗

arXiv2020

New Explicit Lorentzian Einstein-Weyl Structures in 3-Dimensions

On a $3$D manifold, a Weyl geometry consists of pairs $(g, A) =$ (metric, $1$-form) modulo gauge $\widehat{g} = {\rm e}^{2φ} g$, $\widehat{A} = A + {\rm d}φ$. In 1943, Cartan showed that every solution to the Einstein-Weyl equations $R_{(μν)} - \frac{1}{3} R g_{μν} = 0$ comes from an appropriate $3$D leaf space quotient of a $7$D connection bundle associated with a 3$^{\rm rd}$ order ODE $y''' = H(x,y,y',y'')$ modulo point transformations, provided $2$ among $3$ primary point invariants vanish $$ \text{Wünschmann}(H) \equiv 0\equiv \text{Cartan}(H). $$We find that point equivalence of a single PDE $z_y = F(x,y,z,z_x)$ with para-CR integrability $DF := F_x + z_x F_z \equiv 0$ leads to a completely similar $7$D Cartan bundle and connection. Then magically, the (complicated) equation $\text{Wünschmann}(H) \equiv 0$ becomes $$0\equiv\text{Monge}(F):=9F_{pp}^2F_{ppppp}-45F_{pp}F_{ppp}F_{pppp}+40F_{ppp}^3,\qquad p:=z_x, $$ whose solutions are just conics in the $\{p, F\}$-plane. As an ansatz, we take $$F(x,y,z,p):= \frac{α(y)(z-xp)^2+β(y)(z-xp)p+γ(y)(z-xp) +δ(y)p^2+\varepsilon(y)p+ζ(y)}{λ(y)(z-xp)+μ(y) p+ν(y)}, $$ with $9$ arbitrary functions $α, \dots, ν$ of $y$. This $F$ satisfies $DF \equiv 0 \equiv \text{Monge}(F)$, and we show that the condition $\text{Cartan}(H) \equiv 0 $ passes to a certain $\boldsymbol{K}(F) \equiv 0$ which holds for any choice of $α(y), \dots, ν(y)$. Descending to the leaf space quotient, we gain $\infty$-dimensional functionally parametrized and explicit families of Einstein-Weyl structures $\big[ (g, A) \big]$ in $3$D. These structures are nontrivial in the sense that ${\rm d}A \not\equiv 0$ and $\text{Cotton}([g]) \not \equiv 0$.

math.DG↗

arXiv2019

On certain classes of $Sp(2,R)$ symmetric $G_2$ structures

We find two different families of $Sp(2,R)$ symmetric $G_2$ structures in seven dimensions. These are $G_2$ structures with $G_2$ being the split real form of the simple exceptional complex Lie group $G_2$. The first family has $τ_2\equiv 0$, while the second family has $τ_1\equivτ_2\equiv 0$. The families are different in the sense that the first one lives on a homogoneous space $Sp(2,R)/SL(2,R)_l$, and the second one lives on a homogeneous space $Sp(2,R)/Sl(2,R)_s$. Here $SL(2,R)_l$ is an $SL(2,R)$ corresponding to the $\mathfrak{sl}(2,R)$ related to the long roots in the root diagram of $\mathfrak{sp}(2,R)$, and $SL(2,R)_s$ is an $SL(2,R)$ corresponding to the $\mathfrak{sl}(2,R)$ related to the short roots in the root diagram of $\mathfrak{sp}(2,R)$.

math.DG↗

arXiv2019

A car as parabolic geometry

We show that a car, viewed as a nonholonomic system, provides an example of a flat parabolic geometry of type $({\bf SO}(2,3),P_{12})$, where $P_{12}$ is a Borel parabolic subgroup in ${\bf SO}(2,3)$. We discuss the relations of this geometry of a car with the geometry of circles in the plane (a low dimensional Lie sphere geometry), the geometry of 3-dimensional conformal Minkowski spacetime, the geometry of 3-rd order ODEs, projective contact geometry in three dimensions, and the corresponding twistor fibrations. We indicate how all these classical geometries can be interpreted in terms of the nonholonomic kinematics of a car.

math.DG↗

arXiv2000

Intrinsic Geometry of a Null Hypersurface

We apply Cartan's method of equivalence to construct invariants of a given null hypersurface in a Lorentzian space-time. This enables us to fully classify the internal geometry of such surfaces and hence solve the local equivalence problem for null hypersurface structures in 4-dimensional Lorentzian space-times.

gr-qc↗

arXiv2000

Locally Sasakian Manifolds

We show that every Sasakian manifold in dimension $2k+1$ is locally generated by a free real function of $2k$ variables. This function is a Sasakian analogue of the Kähler potential for Kähler geometry. It is also shown that every locally Sasakian-Einstein manifold in $2k+1$ dimensions is generated by a locally Kähler-Einstein manifold in dimension $2k$.

math.DG↗