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I'd like to know the relation between differential geometry and Riemannian geometry.

exactly what makes them different?

Is one an special case of another? exactly how?

moshtaba
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  • Wikipedia article on Riemannian geometry provides a good description: Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point that varies smoothly from point to point. This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other global quantities can be derived by integrating local contributions. – rubikscube09 Jun 04 '18 at 21:31
  • Riemannian geometry is studying a differentiable manifold which is endowed with a Riemannian metric. then is there such a definition for Differential geometry such that it be evident from terms that riamannian geo is an special case of differentiable geo? @rubikscube09 – moshtaba Jun 04 '18 at 21:43
  • Well yes, differential geometry is the study of differentiable manifolds. A differentiable manifold is something (say a surface) that locally ''resembles" $\mathbb{R}^n$ and has a differential structure on it, by which we mean you can define functions on the surface and define their derivatives in a meaningful way. – rubikscube09 Jun 04 '18 at 21:53
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    I think maybe what you say is definition of "differential topology". @rubikscube09 – moshtaba Jun 05 '18 at 01:10

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You can say that

  • Riemannian geometry
  • Lorentz geometry
  • Symplectic geometry
  • Poisson geometry
  • Contact geometry
  • Complex geometry
  • Kähler geometry

and others are instances of differential geometry, which is the study of differentiable manifolds. You have several possibilities of geometric structures to equip your manifold with, and each one gives a "subarea" of differential geometry.

Ivo Terek
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    So what is a "geometric structure" ? is it a privileged tensor field on manifold (as for example in the case of riemannian, symplectic and some others) ? or group structure of lie groups also counted? have we a exact definition for a structure to be "geometric"? (geometric in the sense that we discuss: its presence turns our subject to differential geometry) – moshtaba Jun 05 '18 at 01:21
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    If I recall correctly, there is one general notion of "geometric structure" corresponding to a reduction of the structure of the frame bundle of a manifold from ${\rm GL}(n,\Bbb R)$ to some subgroup of it. For example, a Riemannian metric is a reduction of ${\rm GL}(n,\Bbb R)$ to ${\rm O}(n,\Bbb R)$, a symplectic form is a reduction of ${\rm GL}(2n,\Bbb R)$ to ${\rm Sp}(2n,\Bbb R)$, and so on. But I haven't studied details of that yet. – Ivo Terek Jun 05 '18 at 01:26