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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”.
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 …
- 2,858
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
$$ \ …
- 4,713
8
votes
Accepted
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 …
- 4,713
7
votes
Accepted
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 …
- 34.7k
1
vote
Why trace is more natural than (preferred to) determinant for smooth map $f:M\to N$?
It suffices to understand the special case of a linear map $T:U\to V$ where $U,V$ are Euclidean vector spaces. (Think $U=T_pM$, $V=T_{f(p)}N$, $T=df(p)$.)
Suppose first that $n=\dim V\leq \dim U=m$.
…
- 34.7k
9
votes
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 …
- 34.7k
4
votes
Accepted
Is there a vector field such that one differential form is the Lie derivative of the other?
If $\mu$ is a volume form and $\nu$ is a top degree form, then there exists a vector field $X$ such that $L_X\mu=\nu$ if and only if $\nu$ is exact.
You can always fix a metric $g$ on $M$ such tha …
- 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 …
CommunityBot
- 1
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
3
votes
Can a PDE constrain the degree of a $C^\infty$ map germ?
This question is a bit too general, and I think that it goes beyond the principal symbol. There clearly are constraints. Take for example, the Laplace operator $\Delta$ acting on sections of the triv …
- 4,679
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
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
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
5
votes
Accepted
Higher Cerf Theory
This is what catastrophe theory does, at least for small $n$, $n\leq 10$. Volume 1 of the book by Arnold, Gussein-Zade and Varchenko on singularities has a nice description of this theory; see especi …
- 34.7k
4
votes
Accepted
good reference on brieskorn manifold
I would add Milnor's book "Singular points of complex hypersufaces" Ann of Math Studies, No. 61, Princeton University Press, 1968.
- 34.7k