riemannian-geometry

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    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. S

    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. S

    ##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. S

    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. S

    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. S

    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...
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