Transformation of Quadric Surfaces

Question

Is there a transformation $T: \mathbb{R}^3 \longrightarrow \mathbb{R}^3$ such that a hyperboloid of one-sheet can be mapped to a hyperboloid of two-sheets using such transformation?

Answer 1

Not if you want $T$ to be continuous. There's a theorem that says the continuous image of a connected set is connected. The one sheet hyperboloid has one component, while the two sheet hyperboloid has two components. This means the two sheet hyperboloid is not connected, so no continuout $T$ could map the (connected) one sheet hyberloid to the not-connected two sheet hyperboloic.

More basically, suppose $T$ were continuous, and maps the one-sheet to the two sheet. Take two points $p,q$ on different sheets of the two sheet hyperboloid. Let $a,b$ be the points on the one sheeted hyperboloid for which $T(a)=p$ and $T(b)=q$. [Note that $a,b$ are different points on the one sheet hyperboloid, since otherwise from $a=b$ would follow that $p=T(a)=T(b)=q$, but we chose $p,q$ as points on different sheets of the two sheet hyperboloid, so that we know $p$ and $q$ are different.] Now draw a continuous curve $C$ in the one sheeted hyperboloid connecting $a$ to $b$. Then restrict the map $T$ to this curve, and the image $T(C)$ would connect $p$ to $q$. That's clearly not possible since $p,q$ are on different sheets of the two sheeted hyperboloid.