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Ion I. Dinca

Publications and source records attributed to Ion I. Dinca.

10 recordsLinked to original sources
arXiv2026

On a new definition of the Bäcklund transformation in the isometric deformation of surfaces

We prove that a generic $4$-dimensional integrable rolling distribution of contact elements with the symmetry of the tangency configuration (excluding developable seed and isotropic developable leaves) splits into an $1$-dimensional family of generic $3$-dimensional integrable rolling distributions of contact elements with the symmetry of the tangency configuration, thus introducing a new definition of the Bäcklund transformation in the isometric deformation of surfaces.

math.DG↗
arXiv2024

On isometric correspondence of leaves

We prove that for a generic $3$-dimensional integrable rolling distribution of contact elements (excluding developable seed and isotropic developable leaves) isometric correspondence of leaves of a general nature (independent of the shape of the seed) requires the Bäcklund transformation.

math.DG↗
arXiv2016

On the isometric deformation of surfaces via the Bäcklund transformation

In trying to generalize Bianchi's Bäcklund transformation of quadrics to Bäcklund transformations of isometric deformations of other (classes of) surfaces, we investigate basic features of the isometric deformation of surfaces via the Bäcklund transformation with isometric correspondence of leaves of a general nature (independent of the shape of the seed).

math.DG↗
arXiv2010

Peterson's Deformations of Higher Dimensional Quadrics

We provide the first explicit examples of deformations of higher dimensional quadrics: a straightforward generalization of Peterson's explicit 1-dimensional family of deformations in $\mathbb{C}^3$ of 2-dimensional general quadrics with common conjugate system given by the spherical coordinates on the complex sphere $\mathbb{S}^2\subset\mathbb{C}^3$ to an explicit $(n-1)$-dimensional family of deformations in $\mathbb{C}^{2n-1}$ of $n$-dimensional general quadrics with common conjugate system given by the spherical coordinates on the complex sphere $\mathbb{S}^n\subset\mathbb{C}^{n+1}$ and non-degenerate joined second fundamental forms. It is then proven that this family is maximal.

math.DG↗
arXiv2008

The Bäcklund transforms of Peterson's deformations of quadrics

In trying to provide explicit deformations of quadrics the starting point of our investigation is to use Bianchi's link between real deformations of totally real regions of real paraboloids and various totally real forms of the sine-Gordon equation coupled with Bianchi's simple observation that the vacuum soliton of these totally real forms of the sine-Gordon equation provides precisely Peterson's deformations of such quadrics in order to derive explicit Bäcklund transforms of Peterson's deformations of quadrics. Based also on Bianchi's approach of the Bäcklund transformation for quadrics via common conjugate systems and in analogy to the solitons of the sine-Gordon equation corresponding at the level of the geometric picture to the solitons of the pseudo-sphere we propose a model for the solitons of quadrics.

math.DG↗
arXiv2008

Bianchi's Bäcklund transformation for higher dimensional quadrics

We provide a generalization of Bianchi's Bäcklund transformation from 2-dimensional quadrics to higher dimensional quadrics. The starting point of our investigation is the higher dimensional (infinitesimal) version of Bianchi's main four theorems on the theory of deformations of quadrics and Bianchi's treatment of the Bäcklund transformation for diagonal paraboloids via conjugate systems.

math.DG↗
arXiv2006

The Method of Archimedes in the geometry of quadrics

Confocal quadrics capture (encode) and geometrize spectral properties of symmetric operators. Certain metric-projective properties of confocal quadrics (most of them established in the first half of the XIX$^{\mathrm{th}}$ century) {\it carry out} (stick and transfer) by rolling to and influence surfaces {\it applicable} (isometric) to quadrics and surfaces geometrically linked to these, thus providing a wealth of integrable systems and projective transformations of their solutions. We shall mainly follow Bianchi's discussion of deformations (through bending) of quadrics. Interestingly enough, {\it The Method} of Archimedes (lost for 7 centuries and rediscovered in the same year as Bianchi's discovery (1906), so unknown to Bianchi) applies {\it word by word in both spirit and the letter} and may provide the key to generalizations in other settings.

math.DG↗