conical surface, parametrization, immersion, Gaussian and mean curvatures "Find the parametric form of a conical surface $S$ which is spanned by all rays starting $($but not including $)$ a fixed point $\gamma$ and passing through an arbitrary point on $\gamma$ and passing through an arbitrary point on $\gamma$. Formulate a condition which is necessary and sufficient for $S$ to be immersed. Find the Gaussian and mean curvatures."
My progress so far:
We might as well assume the distinguished point is the origin. Then our surface is parameterized as $$X(s,t) = t\gamma(s),\text{ }t \neq 0,$$$$X_s = \dot{\gamma}(s),$$$$X_t = \gamma(s).$$What do I do next? Could I have a hint?
 A: Going off what you have, $X_t$ and $X_s$ are linearly independent if and only if $\gamma$ and $\dot{\gamma}$ are. This means that $X$ is immersed if and only if the curve is never moving directly toward or away from the distinguished point.
Now, to compute curvature, we will find the eigenvectors of the second fundamental form. Notice that $X$ is invariant under dilations. Thus, the normal vector to $X$ is independent of $t$. It follows that $\gamma(s)$ is a direction of principal curvature $\kappa_1 = 0$. The other direction of principal curvature is the perpendicular direction in the image of $dX$, which is the projection of $\dot{\gamma}(s)$ onto the direction normal to $\gamma(s)$. We may normalize to assume $\|\gamma(s)\| \equiv 1$, $\|\dot{\gamma}(s)\| \equiv 1$ without changing the surface $($why?$)$. Then, $N(s,t) = \dot{\gamma}(s) \times \gamma(s)$, so$${{\partial N}\over{\partial t}} = 0,\text{ and }{{\left\langle {{\partial N}\over{\partial s}}, {{\partial X}\over{\partial s}}\right\rangle}\over{\left\langle {{\partial X}\over{\partial s}}, {{\partial X}\over{\partial s}}\right\rangle}}$$is the other principal curvature. This comes out to$$t\langle \ddot{\gamma}(s) \times \gamma(s),\, \dot{\gamma}(s)\rangle/t^2,\text{ or }(1/t)\det(\dot{\gamma}(s)\text{ }\ddot{\gamma}(s)\text{ }\gamma(s)).$$ So $K = 0$, and $H$ is $1/2$ the above quantity for suitably normalized $\gamma$.
Geometrically, this means that if we had a curve $\delta$ whose velocity vector follows the same path as $\gamma$ up to normalization with $K(\delta) \equiv 1$, the second principal curvature is the torsion of $\delta$ divided by distance to $0$.
A: Well, I guess you compute the first and second fundamental forms:
Here are a couple of terms of the first fundamental form:
$$
g_{ss} = \| \dot{\gamma(s)} \|^2 \\
g_{st} = \dot{\gamma(s)} \cdot \gamma(s) 
...
$$
Clearly $X$ is a generally nice map, but if $X_s$ and $X_t$ are parallel, then the derivative is singular...so it can't be an immersion. In short: avoid curves gamma where $\gamma'(s)$ is a multiple of $\gamma(s)$, for any $s$. 
