  Under certain conditions, a $(1+1)$-dimensional slice $\hat{g}$ of a
spherically symmetric black hole spacetime can be equivariantly embedded in
$(2+1)$-dimensional Minkowski space. The embedding depends on a real parameter
that corresponds physically to the surface gravity $\kappa$ of the black hole
horizon.
  Under conditions that turn out to be closely related, a real surface that
possesses rotational symmetry can be equivariantly embedded in 3-dimensional
Euclidean space. The embedding does not obviously depend on a parameter.
However, the Gaussian curvature is given by a simple formula: If the metric is
written $g = \phi(r)^{-1} dr^2 + \phi(r) d\theta^2$, then
$\K_g=-{1/2}\phi''(r)$.
  This note shows that metrics $g$ and $\hat{g}$ occur in dual pairs, and that
the embeddings described above are orthogonal facets of a single phenomenon. In
particular, the metrics and their respective embeddings differ by a Wick
rotation that preserves the ambient symmetry.
  Consequently, the embedding of $g$ depends on a real parameter. The ambient
space is not smooth, and $\kappa$ is inversely proportional to the cone angle
at the axis of rotation. Further, the Gaussian curvature of $\hat{g}$ is given
by a simple formula that seems not to be widely known.
