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I have studied a basic course in differential geometry and algebraic,differential topology.I have clearly understood the differences between them which is"Differential geometry typically studies Riemannian metrics on manifolds, and properties of them whereas algebraic topology studies algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence",but, I cannot understand similarities between them.I have always wondered how the notion of homotopy,fundamental groups,higher homotopy gps,homology gps etc. work in the setup of riemannian geometry and extrinsic differential geometry which depend completely on notion of riemannian matrices as first one is concerned with classifying manifolds while the other is only concerned with certain local and global properties of manifold.

any help would be appericiated...

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  • $\begingroup$ Topology allows you to keep a notion of distinguishability of points/sets without having a metric to define the distance $\endgroup$ – infinitylord Apr 29 '17 at 13:18
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Well, it frequently occurs in Riemannian Geometry that we need to use ideas from algebraic topology. Without getting into too much detail, here is an example.

One of the basic ideas in Riemannian geometry is the notion of parallel transport of a tangent vector along a smooth curve. Given a point $p\in M$ and a loop $\gamma$ based at $p$, we obtain a diffeomorphism (in fact, an isometry) $\varphi_{\gamma}:T_pM\to T_pM$ defined by taking a tangent vector $v\in T_pM$ and parallel transporting it along $\gamma$. The set $$Hol_p(M):= \{\varphi_{\gamma}: \gamma \ \text{a loop based at} \ p\} $$ of isomorphisms of $T_pM$ arising from parallel transport around loops based at $p$ forms a group under composition, and is called the $\textit{holonomy group based at p}$. Thus, when dealing with holonomy groups it will be highly relevant whether $M$ is simply connected (i.e. whether all loops are contractible).

As another example, there are plenty of theorems in Comparison Geometry which deal with fundamental groups, covering spaces, etc. For example:

$\textbf{Theorem}$ (Preissman): If $M$ is a compact Riemannian manifold of negative curvature, then any non-identity abelian subgroup of $\pi_1(M)$ is infinite cyclic.

Hope this helps!

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