Two places having same gravitational acceleration - Do they also have same curvature of space time? If there are two places (possibly on/above two different planets) having same gravitational acceleration (g), would it imply that the two places have same extent of space time curvature and will have same time dilation? Please answer with minimum required math, no need to put all the math, if lengthy math is involved.
 A: Since you mention planets in the question let's take the Schwarzschild metric that describes the geometry around a spherically symmetric object like (approximately) a planet. The acceleration measured by a stationary observer at a distance $r$ from the planet is:
$$ a = \frac{GM}{r^2} \frac{1}{\sqrt{1-r_s/r}} $$
The time dilation measured by the same observer, relative to an observer at infinity is:
$$ \frac{t_\infty}{\tau} = \frac{1}{\sqrt{1-\frac{r_s}{r}}} $$
So the ratio of the two is:
$$ \frac{a}{t_\infty/\tau} = \frac{GM}{r^2} $$
And this varies with the mass of the object, so it is different for planets with different masses.
Although your question does specifically compare acceleration to time dilation you do also ask a more general question about whether the acceleration can be the same for different curvatures. However this can't be answered without defining what you mean by spacetime curvature. The curvature is described by a matrix (the metric tensor) not a single number so there isn't a simple comparison to make.
