On the thermal stability of a static spherically symmetric black holes in Nash embedding framework

We study the deformation caused by the influence of extrinsic curvature on a vacuum spherically symmetric metric embedded in a five-dimensional bulk. In this sense, we investigate the produced black-holes and derive general characteristics such as their masses, horizons, singularities and thermal properties. As a test, we also study the bending of light near such black-holes analyzing the movement of a test particle and the modification caused by extrinsic curvature on its movement. Accordingly, using the asymptotically conformal flat condition for the extrinsic curvature, an analytical expansion of a set of \emph{n}-scalar fields can be defined and we show that the corresponding black holes must be large and constrained in the range of allowed values $-1/2 \leq n \leq 1.8$. As a result, they are locally thermodynamically stable, but not globally preferred.