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

The B\"{a}cklund transforms of Peterson's isometric deformations of diagonal higher dimensional quadrics without center

We provide the B\"{a}cklund transforms of Peterson's isometric deformations of diagonal higher dimensional quadrics without center. These are found explicitly. can be iterated via the Bianchi Permutability Theorem and can be further iterated via the $3$-M\"{o}bius configuration, thus producing explicit solutions depending on arbitrarily many constants.Comment: arXiv admin note: substantial text overlap with arXiv:0808.2007, arXiv:2208.11419; text overlap with arXiv:0905.0216, arXiv:0902.142