# riemannian-geometry

1. ### I Variation of geometrical quantities under infinitesimal deformation

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...
2. ### Meaning of isomorphism/diffeomorphism $f: R^n\to M^m$

Can one define an isomorphism/diffeomorphism map $f: R^n\to M^m$ when $n>m$? $M$ is a non-compact Riemannian manifold..
3. ### $x+y$ on a Riemannian manifold

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 $M$?
4. ### Taylor expansion of the square of the distance function

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) +...
5. ### Distance function in Riemannian normal coordinates

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}...
6. ### General Relativity and Differential Geometry textbook problem

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...