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Consider the hyperbolic paraboloid $X$ contained in $\mathbb{R}^3$, and the sphere $Y$ in $\mathbb{R}^3$ given by the equations $x^2 −y^2 =z$ and $x^2+y^2+(z−1)2=r^2.$ For what values of $r$ and $b$ is $X ∩ Y$ a smooth manifold? When it is a manifold, what is its dimension?

I know its a manifold when the tangent planes to the points of $X$ and $Y$ span $\mathbb{R}^3$. But the problems seems a bit messy and hard to do.

dunkindonuts
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1 Answers1

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Hint: Consider the map $f:\mathbf R^3\to\mathbf R^2$ defined by $$ f(x,y,z) = (x^2-y^2-z, x^2+y^2+(z-1)^2-r^2).$$ Notice that $X\cap Y=f^{-1}\{(0,0)\}$, and determine for what values of $r$ the map $f$ is a submersion at all points of $X\cap Y$.

jathd
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