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Results tagged with differential-topology
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user 20302
The study of differentiable manifolds and differentiable maps. One fundamental problem is that of classifying manifolds up to diffeomorphism. Differential topology is what Poincaré understood as topology or “analysis situs”.
53
votes
Accepted
Can one recover the smooth Gauss Bonnet theorem from the combinatorial Gauss Bonnet theorem ...
The answer is yes, to both questions.
First question first. For any geodesic $n$-gon $P$ on $M$, i.e., a simply connected region of $M$ whose boundary consists of $n$-geodesic arcs, define
$$ \ …
17
votes
How to get convinced that there are a lot of 3-manifolds?
Here are two examples suggesting the complexity of the world of $3$-manifolds.
The first is the classical result that any $3$-manifold can be obtained by integral surgery on a link in $S^3$. I …
- 34.7k
16
votes
Sheaf-theoretically characterize a Riemannian structure?
Suppose that $M$ is a smooth manifold and $g_0, g_1$ are Riemann metrics on $M$. $\newcommand{\eH}{\mathscr{H}}$ Denote by $\eH_{g_i}$, $i=0,1$ and the sheaf of $g_i$-harmonic functions. More precise …
- 34.7k
15
votes
Accepted
Measures and differential forms on manifolds
I assume that $\mu$ is a measure defined on the $\sigma$-algebra of Borel sets. First, on any manifold the notion of negligible set is well defined.
If $M$ is orientable and $\mu(N)=0$ for any neglig …
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14
votes
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The difference between a handle decomposition and a CW decomposition
The second of the theorems you quoted is considerably harder to prove. The gist of the proof is as follows. Consider the closure $\overline{D(p)}$ of $D(p)$ in $M$. Then Lizhen Qin proves that it …
- 34.7k
10
votes
On the generalized Gauss-Bonnet theorem
You can try Chap. 13, vol. 5 of M. Spivak's opus A comprehensive Introduction to Differential Geometry. (On the cover of this volume there are three birds carrying a banner that reads "All the …
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10
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Where can I find a full proof of the Chern-Gauss-Bonnet theorem ?
For a complete proof of the Gauss-Bonnet-Chern for arbitrary vector bundles (not just tangent bundles) see Section 8.3.2 of these notes. The proof is Chern's original proof, based on Chern-Weil the …
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10
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Floer homology and Invariants for Einstein Field Equations?
Here is one problem. The instantons or the monopoles are critical points of certain energy functionals and thus they satisfy Euler-Lagrange equations. These are second-order p.d.e.'s. However the …
- 34.7k
9
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Smooth Morse function from Forman's discrete Morse function
You can do the next best think. To a Forman-Morse function $f$ one can associate a flow on the manifold whose stationary points are precisely the barycenters of the faces of your simplicial decom …
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8
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A description of cellular boundary maps in terms of a Morse function
Under certain conditions, (Morse-Smale being one, but not sufficient) the stratification by unstable manifolds of a Morse flow on a compact manifolds gives a cellular decomposition; see the paper On …
- 34.7k
7
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Proving the existence of good covers
The answer to both questions is yes. Fix a triangulation of the manifold. For any vertex $v$ denote by $U_v$ the union of the relative interiors of all the faces of all dimensions that contain the ve …
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7
votes
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The purpose of connections in differential geometry
If you are interested in local-to-global results, i.e., collecting local info about the manifold and then patch it together to get a global info then you need tools for the patching part of the proces …
7
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Line bundle on $S^2$
Along the same lines. Rank $1$ real vector bundles over a compact CW complex $X$ are all pullbacks of the tautological line bundle over $\mathbb{RP}^\infty$. The space of isomorphism classes of su …
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7
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Is there a natural form representing the Thom class of a vector bundle, which when pulled ba...
Here is another construction which goes back to Chern's proof of the Gauss-Bonnet theorem.
Suppose that $\pi: E\to M$ is an oriented rank $2k$ real vector bundle over the manifold $M$. Assume ad …
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6
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Reference request: an elementary proof of Brouwer fixed-point theorem.
There is a completely elementary and very elegant proof of the Brower fixed point theorem based on a beautiful combinatorial result called Sperner lemma. For details I recommend Section 2.3, page 7 …
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