arXiv subjects

Josef Šilhan

Publications and source records attributed to Josef Šilhan.

12 recordsLinked to original sources

arXiv2026

Compatibility complexes for the conformal-to-Einstein operator

The conformal-to-Einstein operator is a conformally invariant linear overdetermined differential operator whose non-vanishing solutions correspond to Einstein metrics within a conformal class. We construct compatibility complexes for this operator under natural genericity assumptions on the Weyl curvature in dimension $n\ge 4$, which implies at most one independent solution. An analogous result for the projective-to-Ricci-flat operator is obtained as well. The construction is based on a method, previously proposed by one of the authors, that leverages existing symmetries and geometric properties of the starting operator. In this case the compatibility complexes consist of, respectively, conformally and projectively invariant operators. We also make some comments on how Bernstein-Gelfand-Gelfand sequences can be interpreted as compatibility complexes in the locally flat case, which may be of general interest.

math.DG↗

arXiv2025

Holography of Higher Codimension Submanifolds: Riemannian and Conformal

We provide a natural generalization to submanifolds of the holographic method used to extract higher-order local invariants of both Riemannian and conformal embeddings, some of which depend on a choice of parallelization of the normal bundle. Qualitatively new behavior is observed in the higher-codimension case, giving rise to new invariants that obstruct the order-by-order construction of unit defining maps. In the conformal setting, a novel invariant (that vanishes in codimension 1) is realized as the leading transverse-order term appearing in a holographically-constructed Willmore invariant. Using these same tools, we also investigate the formal solutions to extension problems off of an embedded submanifold.

math.DG↗

arXiv2024

Inverting covariant exterior derivative

The algorithm for inverting covariant exterior derivative is provided. It works for a sufficiently small star-shaped region of a fibered set - a local subset of a vector bundle and associated vector bundle. The algorithm contains some constraints that can fail, giving no solution, which is the expected case for parallel transport equations. These constraints are straightforward to obtain in the proposed approach. The relation to operational calculus and operator theory is outlined. The upshot of this paper is to show, using the linear homotopy operator of the Poincare lemma, that we can solve the covariant constant and related equations in a geometric and algorithmic way. The considerations related to the regularity of the solutions are provided.

math.DG↗

arXiv2024

The gap phenomenon for conformally related Einstein metrics

We determine the submaximal dimensions of the spaces of almost Einstein scales and normal conformal Killing fields for connected conformal manifolds. The results depend on the signature and dimension $n$ of the conformally nonflat conformal manifold. In the Riemannian case, these two dimensions are at most $n-3$ and $\frac{(n-4)(n-3)}{2}$, respectively. In the Lorentzian case, these two dimensions are at most $n-2$ and $\frac{(n-3)(n-2)}{2}$, respectively. In the remaining signatures, these two dimensions are at most $n-1$ and $\frac{(n-2)(n-1)}{2}$, respectively. This upper bound is sharp and to realize examples of submaximal dimensions, we first provide them directly in dimension 4. In higher dimensions, we construct the submaximal examples as the (warped) product of the (pseudo)-Euclidean base of dimension $n-4$ with one of the 4-dimensional submaximal examples.

math.DG↗

arXiv2023

Modified conformal extensions

We present a geometric construction and characterization of $2n$-dimensional split-signature conformal structures endowed with a twistor spinor with integrable kernel. The construction is regarded as a modification of the conformal Patterson--Walker metric construction for $n$-dimensional projective manifolds. The characterization is presented in terms of the twistor spinor and an integrability condition on the conformal Weyl curvature. We further derive complete description of Einstein metrics and infinitesimal conformal symmetries in terms of suitable projective data. Finally, we obtain an explicit geometrically constructed Fefferman--Graham ambient metric and show vanishing of $Q$-curvature.

math.DG↗

arXiv2020

Conformal Patterson-Walker metrics

The classical Patterson-Walker construction of a split-signature (pseudo-)Riemannian structure from a given torsion-free affine connection is generalized to a construction of a split-signature conformal structure from a given projective class of connections. A characterization of the induced structures is obtained. We achieve a complete description of Einstein metrics in the conformal class formed by the Patterson-Walker metric. Finally, we describe all symmetries of the conformal Patterson-Walker metric. In both cases we obtain descriptions in terms of geometric data on the original structure.

math.DG↗

arXiv2018

Conformal theory of curves with tractors

We present the general theory of curves in conformal geometry using tractor calculus. This primarily involves a tractorial determination of distinguished parametrizations and relative and absolute conformal invariants of generic curves. The absolute conformal invariants are defined via a tractor analogue of the classical Frenet frame construction and then expressed in terms of relative ones. This approach applies likewise to conformal structures of any signature; in the case of indefinite signature we focus especially on the null curves. It also provides a conceptual tool for handling distinguished families of curves (conformal circles and conformal null helices) and conserved quantities along them.

math.DG↗

arXiv2018

Higher symmetries of symplectic Dirac operator

We construct in projective differential geometry of the real dimension $2$ higher symmetry algebra of the symplectic Dirac operator ${D}\kern-0.5em\raise0.22ex\hbox{/}_s$ acting on symplectic spinors. The higher symmetry differential operators correspond to the solution space of a class of projectively invariant overdetermined operators of arbitrarily high order acting on symmetric tensors. The higher symmetry algebra structure corresponds to a completely prime primitive ideal having as its associated variety the minimal nilpotent orbit of $\mathfrak{sl}(3,{\mathbb{R}})$.

math.DG↗

arXiv2017

A Projective-to-Conformal Fefferman-Type Construction

We study a Fefferman-type construction based on the inclusion of Lie groups ${\rm SL}(n+1)$ into ${\rm Spin}(n+1,n+1)$. The construction associates a split-signature $(n,n)$-conformal spin structure to a projective structure of dimension $n$. We prove the existence of a canonical pure twistor spinor and a light-like conformal Killing field on the constructed conformal space. We obtain a complete characterisation of the constructed conformal spaces in terms of these solutions to overdetermined equations and an integrability condition on the Weyl curvature. The Fefferman-type construction presented here can be understood as an alternative approach to study a conformal version of classical Patterson-Walker metrics as discussed in recent works by Dunajski-Tod and by the authors. The present work therefore gives a complete exposition of conformal Patterson-Walker metrics from the viewpoint of parabolic geometry.

math.DG↗

arXiv2016

Fefferman-Graham ambient metrics of Patterson-Walker metrics

Given an $n$-dimensional manifold $N$ with an affine connection $D$, we show that the associated Patterson-Walker metric $g$ on $T^*N$ admits a global and explicit Fefferman-Graham ambient metric. This provides a new and large class of conformal structures which are generically not conformally Einstein but for which the ambient metric exists to all orders and can be realized in a natural and explicit way. In particular, it follows that Patterson-Walker metrics have vanishing Fefferman-Graham obstruction tensors. As an application of the concrete ambient metric realization we show in addition that Patterson-Walker metrics have vanishing Q-curvature.

math.DG↗

arXiv2014

Second Order Symmetries of the Conformal Laplacian

Let (M,g) be an arbitrary pseudo-Riemannian manifold of dimension at least 3. We determine the form of all the conformal symmetries of the conformal (or Yamabe) Laplacian on (M,g), which are given by differential operators of second order. They are constructed from conformal Killing 2-tensors satisfying a natural and conformally invariant condition. As a consequence, we get also the classification of the second order symmetries of the conformal Laplacian. Our results generalize the ones of Eastwood and Carter, which hold on conformally flat and Einstein manifolds respectively. We illustrate our results on two families of examples in dimension three.

math-ph↗

arXiv2011

Invariant prolongation of overdetermined PDE's in projective, conformal and Grassmannian geometry

This is the second in a series of papers on natural modification of the normal tractor connection in a parabolic geometry, which naturally prolongs an underlying overdetermined system of invariant differential equations. We give a short review of the general procedure developed in [5] and then compute the prolongation covariant derivatives for a number of interesting examples in projective, conformal and Grassmannian geometries.

math.DG↗