For which values of $a$ does the hyperboloid $x^2+y^2-z^2=1$ intersect the sphere $x^2+y^2+z^2=a$ transversally? What does the intersection look like for different values of $a$?
What I can see is that if $\sqrt a<1, $ then the intersection is empty and hence transverse. If $\sqrt a=1\iff a=1$, then the intersection points lie on the circle $x^2+y^2=1$ on the $z=0$ plane. I guess at those points the tangent spaces to the hyperboloid and the sphere coincide (they are 2-dim planes). Is that correct? How do I show it more rigorously? (I can use local parametrizations of those manifolds and then compute the image of the corresponding differentials, but this seems to be a huge hassle.) As for the case $\sqrt a > 1$, I guess here the intersection is transverse, but again, how do I show it?