I am studying Peterson's book of riemannian geometry and he gives me a metric:
$$g = dt^2 + a^2t^2d\theta^2$$
and asks me to identify which are the spaces when I change $a$.
I never expected anything like this before, how can I think about this problem?
If $0 < a$, the metric $$ g = dt^{2} + a^{2} t^{2}\, d\theta^{2},\qquad 0 < t,\quad 0 \leq \theta \leq 2\pi, $$ is flat, and represents a cone if $0 < a < 1$, a flat plane if $a = 1$, or a "saddle cone" (non-standard term?) if $1 < a$.
Each such metric embeds isometrically in Euclidean $3$-space, and the space of isometric embeddings is infinite-dimensional: Pick a smooth, embedded, constant-speed curve $\gamma$ of length $2\pi a$ on the unit sphere, and define $$ \phi(t, \theta) = t\gamma(\theta). $$ Since $\phi_{t} = \gamma$ lies on the unit sphere and $\phi_{\theta} = t\gamma'$ is tangent to the sphere and has constant speed $a$, the components of the induced metric are $$ E = 1,\qquad F = 0,\qquad G = t^{2} \|\gamma'\|^{2} = a^{2} t^{2}. $$