Consider the cartesian product $\mathbb{H}_{n}$ of $n$ $m-1$-dimensional forward hyperbolae in $\mathbb{R}^{mn}$ as given by the parametrization:

$\mathbb{H}_i: \ \ x_i=\sqrt{(\vec{x}_{i+1}^2+1)}, \hspace{1cm} i=1,3,...,2n-1, \ (x_i,\vec{x}_{i+1})\in\mathbb{R}^{m}$

and consider now the intersection of $\mathbb{H}_{n}$ with a backward hyperboloid (rather, when thought in $\mathbb{R}^{mn}$, an hyperbolic cylinder), given by the equation

$\mathbb{H}': \ \ c+\sum_{j \ \text{odd}} A_j x_j=-\sqrt{\Big(\sum_{j \ \text{even}} A_j \vec{x}_j \Big)^2 +1 } $

where $A_j=A_{j+1}$ and $A_j\in\{-1,0,1\}$ and $c\in\mathbb{R}$.

The first question is: what is the nature of this intersection? From what I can understand, when restricted to the subspace $(x_i,\vec{x}_{i+1})\in\mathbb{R}^m$, this intersection is $\mathbb{H}_i$ if $A_i=0$, a lower dimensional hyperboloid if $A_i=-1$ and an ellipse if $A_i=1$, provided that $c<-1$. In case $A_i=\pm 1$ the position of the foci of the ellipses and hyperboloids might depend on the value of the other $(x_j,\vec{x}_{j+1}) \ j\ne i$. Is there a name for this kind of manifolds?

The second question is: what can one say of the image of this intersection by an affine map (whose linear part is an isomorphism)? I would espect that the general structure in preserved, since an ellipse is mapped in an ellipse by an affine map and the same holds for an hyperboloid.

  • $\begingroup$ How is defined a "forward hyperbola" ? $\endgroup$ – Jean Marie Dec 27 '18 at 15:20
  • $\begingroup$ Given the implicit equation, e.g in two dimensions, $x^2-y^2=1$, one can solve for x, obtaining two sheets, i.e. $x=\pm\sqrt{1+y^2}$. Then $x=\sqrt{1+y^2}$ is the forward hyperbola and $x=-\sqrt{1+y^2}$ the backward hyperbola. $\endgroup$ – Tanatofobico Dec 27 '18 at 15:23
  • $\begingroup$ OK, the branch of hyperbola situated in the first quadrant. $\endgroup$ – Jean Marie Dec 27 '18 at 15:27

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