This question is about 2-d surfaces embedded inR3
It's easy to find information on how the metric tensor changes when $$x_{\mu}\rightarrow x_{\mu}+\varepsilon\xi(x)$$
So, what about the variation of the second fundamental form, the Gauss and the mean curvature? how they change?
I found...
Can one define a vector space structure on a Riemannian manifold ##(M,g)##?! By this I mean, does it make a sense to write ##x+y## where ##x,y## are arbitrary points on ##M##?
Does it make a sense to define the Taylor expansion of the square of the distance function? If so, how can one compute its coefficients? I simply thought that the square of the distance function is a scalar function, so I think that one can write
$$
d^2(x,x_0)=d^2(x'+(x-x'),x_0)=d^2(x',x_0) +...
Hi,
I read somewhere the geodesic distance between an arbitrary point ##x## and the base point ##x_0## in normal coordinates is just the Euclidean distance. Why?! That's the part I don't understand. I know that one can write
g_{\mu \nu} = \delta_{\mu \nu} - \frac{1}{6} (R_{\mu \rho \nu \sigma}...
I'm studying General Relativity and Differential Geometry. In my text book, the author has written ##x^2=d(x,.)## where d(x,y) is distance between two points ##x,y\in M##. I couldn't understand what d(x,.) means. Moreover, I am not sure if this is generally true to write ##x^2=g_{\mu\nu} x^\mu...