Second-Order Surfaces

Let $f_{2}(x)+2 f_{1}(x)+f_{0} = 0$ be the general equation of a second-order surface in the affine space $\mathbb {R}^{n}$, where $f_{2}$ and $f_{1}$ are a quadratic and a linear function respectively, and $f_{0}$ is a number, and let $r \geqslant 1$ be the rank of the quadratic function $f_{2}$. Prove that the following list of equations gives the affine classification of second-order surfaces in $\mathbb {R}^{n}$. I. $x_{1}^{2}+\ldots +x_{k}^{2}-x_{k+1}^{2}-\ldots -x_{r}^{2} = 1,0 \leqslant k \leqslant r$.

For $k = r = n$ the surface is called a real $(n-1)$-dimensional ellipsoid; for $k = 0, r = n$ an imaginary $(n-1)$-dimensional ellipsoid; for $0 < k < r = n$ it is called an $(n-1)$-dimensional hyperboloid; for $r < n$ a cylinder over the corresponding $(r-1)$-dimensional surface.

II. $x_{1}^{2}+\ldots +x_{k}^{2}-x_{k+1}^{2}-\ldots -x_{r}^{2} = 0, k \geqslant \frac{r}{2}$ for even $r$ and $k \geqslant \frac{r+1}{2}$ for odd $r$.

For $k < r = n$ the surface is called a real $(n-1)$-dimensional cone; for $r < n$ a cylinder over an $(r-1)$-dimensional cone, or a conical surface.

III. $x_{1}^{2}+\ldots +x_{k}^{2}-x_{k+1}^{2}-\ldots -x_{r}^{2} = 2 x_{r+1}, k \geqslant \frac{r}{2}$ for even $r < n$ and $k \geqslant \frac{r+1}{2}$ for odd $r < n$.

For $r = n-1$ the surface is called an $(n-1)$-dimensional paraboloid; for $r < n-1$ a cylinder over the corresponding $(r-1)$-dimensional paraboloid.

Determine the affine type of the section of the surface

by the plane $x_{1}+x_{2}+x_{3}+x_{4}-1 = 0$.

Find the canonical equation and the canonical coordinate system for the surface:

Determine the affine type of the section of the surface

by the plane $x_{2}+x_{3}+x_{4}-1 = 0$.

Find the maximal dimension of a plane contained in a second-order surface given in an $n$-dimensional affine space, if this surface is:

Find all rectilinear generators of the surface $x_{1}^{2} + x_{2}^{2}-x_{3}^{2} = 2 x_{4}$, passing through the point $(1,1,1,1 / 2)$.

Let $f\left(x_{1}, \ldots , x_{n}\right)$ and $g\left(x_{1}, \ldots , x_{n}\right)$ be two quadratic functions, at least one of which is positive definite. Prove that the surfaces $f = 1$ and $g = 1$ do not intersect if and only if the function $f-g$ is sign-definite.