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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”.
7
votes
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 …
- 34.7k
1
vote
closed dual of vector fields
Fix a Riemann metric metric $g$ on the manifold. Denote by $\omega$ the $1$-form dual to $X$ defined by
$$\omega(Y)= g(X,Y), $$
for any vector field $Y$. The $1$-form
$$\alpha :=\frac{1}{|X|^2_g} …
- 34.7k
0
votes
Tubular neighborhoods of chains
Have you looked at the deformation theorem for rectifiable currents? This essentially states that any integral current $S$ can be approximated by a polyhedral current situated not very far from $ …
- 34.7k
10
votes
On the generalized Gauss-Bonnet theorem
You can try Chap. 13, vol. 5 of M. Spivak's opus A comprehensive Introduction to Differential Geometry. (On the cover of this volume there are three birds carrying a banner that reads "All the …
- 34.7k
5
votes
Most general context for the Morse Lemmas
The questions that you asked are addressed by the once very sexy field of catastrophe theory. The story is a bit too long to tell here. The conditions you are asking for are called stability co …
- 34.7k
3
votes
Euler class in the non-compact case
There is one version of Euler class for oriented vector bundles on non-compact manifolds, the so called relative Euler class . It requires that the vector bundle admits a section which does not vani …
- 34.7k
15
votes
Accepted
Measures and differential forms on manifolds
I assume that $\mu$ is a measure defined on the $\sigma$-algebra of Borel sets. First, on any manifold the notion of negligible set is well defined.
If $M$ is orientable and $\mu(N)=0$ for any neglig …
- 34.7k
2
votes
Classification of natural invariants of Riemannian structures
There are several of those appearing in the definition of the so-called Quermassintegrals. On a manifold of dimension $m$ there are $\lfloor m/2\rfloor+1$ such integrals
$$Q_m(M) =\int_M |dV_g|,\;\ …
- 34.7k
4
votes
Accepted
dimension of a union of grassmannians
Denote by $S^{d-1}$ the unit sphere in $\newcommand{\bR}{\mathbb{R}}$ $\bR^d$ and consider the manifold
$$ X= \bigl\lbrace (v,t)\in S^{d-1}\times \bR;\;\; v\perp \gamma(t)\;\bigr\rbrace. $$
The …
- 34.7k
6
votes
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 …
- 34.7k
2
votes
Computation of the Euler characteristic of a specific real variety
This is tricky even in the simplest case. Suppose we are given a real polynomial in one real variable. The Euler characteristic of its zero set is equal to the number of real roots (not counted wit …
- 34.7k
4
votes
Map between manifolds and open dense subsets
The answer is positive if we can assume that $U$ is Borel measurable (not necessarily open) and its complement is negligible. I will assume this in the sequel.
Fix Riemann metrics $g$ on $X$ and …
- 34.7k
7
votes
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 …
- 34.7k
14
votes
Accepted
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
1
vote
Tensor bundles as G structures
The group $GL(n,\mathbb{R})$ does not act by conjugation on symmetric bilinear forms. If $A$ is the symmetric $n\times n$ matrix describing one such form in a given basis and $S$ is a linear invertib …
- 34.7k