Adam Chudecki

A list below contains the problems which I am dealing with nowadays.

1. Exact and general solutions of para-Kähler spaces. This issue deals with spaces of the types $[\textrm{any}] \otimes [\textrm{D}]^{nn}$. Especially interesting are Einstein spaces of the types $[\textrm{I}] \otimes [\textrm{D}]^{nn}$ because no explicit examples of such spaces are known nowadays. Also, new examples of the heavenly spaces of the type $[\textrm{I}] \otimes [\textrm{O}]$ could be found during the analysis of this problem.
2. Para-Kähler Einstein geometry of the neutral spaces of the types $[\textrm{D}]\otimes [\textrm{N}]$. Cosmological constant $\Lambda$ is necessarily nonzero in this case. There are two subtypes: $\{ [\textrm{D}]^{nn} \otimes [\textrm{N}]^{e},[--,++] \}$ and $\{ [\textrm{D}]^{nn} \otimes [\textrm{N}]^{e},[++,++] \}$. It is supposed that the first of these types could be solved with all the generality. Interesting task is to find examples of the second type.
3. Para-Hermite Einstein geometry of the neutral spaces of the types $[\textrm{D}]\otimes [\textrm{N}]$. There are two subtypes: $[\textrm{D}]^{ee} \otimes [\textrm{N}]^{n}$ and $[\textrm{D}]^{ee} \otimes [\textrm{N}]^{e}$ with different properties of the congruences of null geodesics. Explicit examples are unknown.
   
4. Spaces equipped with three or four distinct congruences of null strings of the same duality. The metric of a space equipped with three congruences of null strings is known - this is a space of the type $[\textrm{I,II}]^{eee} \otimes [\textrm{any}]$ (with a special subcase $[\textrm{II}]^{nee} \otimes [\textrm{any}]$). However, the metric of a space of the type $[\textrm{I}]^{eeee} \otimes [\textrm{any}]$ remains unknown. Can the results be specialized to the types $[\textrm{I}]^{eeee} \otimes [\textrm{D}]^{nn}$ and $[\textrm{I}]^{eeee} \otimes [\textrm{O}]$ (which are para-Kähler)? What types of the traceless Ricci tensor are admitted by such spaces?
5. Spaces equipped with four distinct congruences of null geodesics. The problem is to find the metric of a space which is equipped with four distinct shearfree congruences of null geodesics. It is not even known if such a structure exists. If "yes", it is admitted only by spaces of the type $[\textrm{I}]$ with matter. What types of mater are admitted by such spaces?
6. Killing vectors in Walker spaces in dimension 4. The main aim is to analyze the Killing symmetries in spaces of the types $[\textrm{deg}]^{n} \otimes [\textrm{any}]$. Is there an equivalent of the master equation in such spaces?
7. Null Killing vectors in spaces with $C_{ab} \ne 0$. It is known (A.C., 2014) that the existence of the null Killing vectors in Einstein spaces implies the existence of the self-dual and anti-self-dual congruences of null strings. Can these results be generalized to the case of non-vacuum spaces.
8. Type-$[\textrm{N}] \otimes [\textrm{N}]$ problem - a superposition. It is known (Plebański, Przanowski, Formański, 1998) that a superposition of two heavenly spaces of the types $[\textrm{N}] \otimes [\textrm{O}]$ and $[\textrm{O}] \otimes [\textrm{N}]$ lead to the space of the type $[\textrm{N}] \otimes [\textrm{N}]$ with matter. The idea is to consider superposition of two SD spaces of the types $[\textrm{N}] \otimes [\textrm{O}]$ and $[\textrm{O}] \otimes [\textrm{N}]$? The general metric of the SD space of the type $[\textrm{N}]^{n} \otimes [\textrm{O}]^{n}$ is known. It is desired to find a general form of the metric of the type $[\textrm{N}]^{e} \otimes [\textrm{O}]^{n}$ and then investigate a superposition.
9. Type-$[\textrm{N}] \otimes [\textrm{N}]$ spaces with a single symmetry. Is it possible to reduce the vacuum Einstein field equations for the type $[\textrm{N}]^{e} \otimes [\textrm{N}]^{e}$ with a twist equipped with a single Killing vector to a single equation? Can the counterpart of the Hauser solution be found?
10. Type-$[\textrm{N}] \otimes [\textrm{N}]$ spaces with $\Lambda \ne 0$. It is known that the existence of the nonzero cosmological constant simplifies a little type-$[\textrm{N}]$ problem with twist. With $\Lambda \ne 0$ there are only three subtypes, namely $\{ [\textrm{N}]^{e} \otimes [\textrm{N}]^{e},[--] \}$, $\{ [\textrm{N}]^{e} \otimes [\textrm{N}]^{e},[+-] \}$ and $\{ [\textrm{N}]^{e} \otimes [\textrm{N}]^{e},[++] \}$. It is possible to reconstruct known solutions and to find new ones?

Issues 1-4 and 6 deal with neutral geometry (in particular: para-Kähler and para-Hermitian geometry). Issues 8-10 are related to the almost 50-years old problem, namely: to find the second twisting, vacuum type $[\textrm{N}]$ solution. In all these problems a complex methods and spinorial formalism seem to be the best tools for analysis.

A title of the Ph.D. thesis which can be proposed is "Hyperheavenly spaces and their real slices". For realization of this dissertation a basic knowledge of differential geometry, general theory of relativity and partial differential equations is needed. The dissertation can be finalized in 4-5 years.

Journal articles:

• Hyperheavenly spaces and their application in Walker and para-Kähler geometries: Part II, Journal of Geometry and Physics 188, 104826 (2023); arXiv, link
• Hyperheavenly spaces and their application in Walker and para-Kähler geometries: Part I, Journal of Geometry and Physics 179, 104591 (2022); arXiv, link
• Two-sided conformally recurrent self-dual spaces, Journal of Geometry and Physics 159, 103933 (2021); arXiv; link
• Classification of complex and real vacuum spaces of the type $[\textrm{N}] \otimes [\textrm{N}]$, Journal of Mathematical Physics 59, 062503 (2018); arXiv; link
• On twisting type $[\textrm{N}] \otimes [\textrm{N}]$ Ricci flat complex spacetimes with two homothetic symmetries, Journal of Mathematical Physics 59, 042504 (2018) (współautor: M. Przanowski); arXiv; link
• On geometry of congruences of null strings in 4-dimensional complex and real pseudo-Riemannian spaces, Journal of Mathematical Physics 58, 112502 (2017); arXiv; link
• On some examples of para-Hermite and para-Kähler Einstein spaces with $\Lambda \ne 0$, Journal of Geometry and Physics 112, 175–196 (2017); arXiv; link
• Classification of the traceless Ricci tensor in 4-dimensional pseudo-Riemannian spaces of neutral signature, Acta Physica Polonica B, Vol. 48, No. 1, 53-74 (2017); arXiv; link
• All complex and real ASD Einstein spaces with $\Lambda$ admitting nonnull Killing vector, International Journal of Geometric Methods in Modern Physics, Vol. 13, No. 2, 1650011 (2016); arXiv; link
• Proper conformal symmetries in self-dual Einstein spaces, Journal of Mathematical Physics 55, 082502 (2014) (współautor: M. Dobrski); arXiv; link
• Null Killing vectors and geometry of null strings in Einstein Spaces, General Relativity and Gravitation 46, 1714 (2014); arXiv; link
• Killing Symmetries in $\mathcal{H}$ spaces with $\Lambda$, Journal of Mathematical Physics 54, 102503 (2013) (współautor: M. Przanowski); arXiv; link
• Homothetic Killing vectors in expanding $\mathcal{HH}$-spaces with $\Lambda$, International Journal of Geometric Methods in Modern Physics, Vol. 10, No. 1, 1250077 (2013); arXiv; link
• Classification of the Killing vectors in nonexpanding $\mathcal{HH}$-spaces with $\Lambda$, Classical and Quantum Gravity 29, 135010 (2012); arXiv; link
• Notes on para-Hermite-Einstein spacetimes, International Journal of Geometric Methods in Modern Physics, Vol. 9, No. 1, 1250008 (2012) (współautorzy: M. Przanowski i S. Formański); link
• Conformal Killing vectors in nonexpanding $\mathcal{HH}$-spaces with $\Lambda$, Classical and Quantum Gravity 27, 205004 (2010); link
• From hyperheavenly spaces to Walker and Osserman spaces: II, Classical and Quantum Gravity 25, 235019 (2008) (współautor: M. Przanowski); link
• From hyperheavenly spaces to Walker and Osserman spaces: I, Classical and Quantum Gravity 25, 145010 (2008) (współautor: M. Przanowski); link
• A simple example of type-$[\textrm{N}]\otimes[\textrm{N}]$ $\mathcal{HH}$-spaces admitting twisting null geodesic congruence, Classical and Quantum Gravity 25, 055010 (2008) (współautor: M. Przanowski); link

Conference articles:

• Complex and real para-Kähler Einstein spaces, Acta Physica Polonica B Proceedings Supplement, Vol. 16, No. 6 (2023); link
• Two-sided Walker and para-Kähler Spaces and Real Slices of Hyperheavenly Spaces, Acta Physica Polonica B Proceedings Supplement, Vol. 15, No. 1 (2022); link
• On Some Solutions of the Type [D] Self-dual Spaces, Acta Physica Polonica B Proceedings Supplement, Vol. 13, No. 2 (2020); link
• Congruences of null strings and their relations with Weyl tensor and traceless Ricci tensor, Acta Physica Polonica B Proceedings Supplement, Vol. 10, No. 2 (2017); link