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Results tagged with differential-topology
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user 4463
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”.
41
votes
3
answers
3k
views
Can one recover the smooth Gauss Bonnet theorem from the combinatorial Gauss Bonnet theorem ...
First let me state two known theorems.
Theorem 1 (for smooth manifolds): Let $(M,g)$ be a smooth compact two dimensional Riemannian manifold. Then
$$ \int \frac{K}{2 \pi} dA = \chi (M) $$
where $K$ i …
- 3,245
16
votes
5
answers
1k
views
Take contraction wrt a vector field twice and define kernel mod image. Does that give anythi...
First, we make the following observation: let $X: M \rightarrow TM $ be a vector
field on a smooth manifold. Taking the contraction with respect to $X$ twice gives zero, i.e.
$$ i_X \circ i_{X} =0.$ …
- 3,245
9
votes
1
answer
935
views
Do partitions of unity exist if we impose additional conditions on the derivatives?
Let $ ~~\cup_{k=-1}^{\infty} U_k = \mathbb{R} $ be an open covering of
$\mathbb{R}$. It is a well known fact that partitions of unity subbordinate to
the cover exists, i.e. there exists smooth
func …
- 3,245
9
votes
3
answers
2k
views
Is there a natural form representing the Thom class of a vector bundle, which when pulled ba...
Let $V \rightarrow M$ be an oriented vector bundle over a compact
oriented manifold $M$ equipped with a metric $h$ (the metric $h$
is a metric on the Vector bundle $V$, not on the manifold $M$).
Is …
- 3,245
8
votes
2
answers
414
views
Are there analogous statements for the number of zeros of a section in terms of the Euler cl...
Let $V \rightarrow M$ be an oriented rank $k$ vector bundle over a compact orientd manifold $M$. Let $X \subset M$ be a
compact topological subspace of $M$ that is a smooth oriented submanifold of
d …
- 3,245
5
votes
1
answer
365
views
Is the space of real conics with a singular point an orientable manifold?
Consider the space of non zero real homogeneous degree $2$ polynomials in three variables upto scaling. This space is $\mathbb{R} \mathbb{P}^5$. The zero set
of such a polynomial gives a real curve i …
- 3,245
4
votes
2
answers
553
views
Is there an analogous concept for the degree of a map, when the spaces are singular?
Let $M$ and $N$ be two smooth compact, oriented manifolds and
$X\subset M$ an oriented submanifold of $M$ of dimension $k$
(not necessarily closed). Suppose in addition that $\bar{X}-X$ is contained …
- 3,245
4
votes
2
answers
413
views
Is it impossible for the dimension of a topological space to increase under a smooth map?
First let me make a definition. Let $M$ be a smooth manifold and
$S \subset M $ a topological subspace of $M$. We say that $S$ has
"dimenion" at most $k$ if $S$ is a subset of
$$ X_1 \cup X_2 \ldots …
- 3,245
4
votes
3
answers
562
views
What is the easiest way to show that three lines in two dimensional space do not intersect?
I have two similar questions:
1) Let $X$ and $Y$ be two measure spaces. Suppose for every point
$x \in X $ there exists a set $ \mathcal{U}_x \subset Y $ of full
measure in $Y$. Suppose $V \subset …
- 3,245
3
votes
1
answer
214
views
Is there a formula for the intersection of projectivized lines inside a projectivized vector...
Let $E\rightarrow D$ be a complex rank two vector bundle over a compact complex
one dimensional manifold $D$. Let $L_1, L_2 \subset E$ be rank one subbundles of E
(i.e. line bundles). Let
$$ n_1:= …
- 3,245
3
votes
1
answer
360
views
General position argument for reasonable spaces
Let $\mathcal{D} \approx \mathbb{P}^{\delta_d}$ be the space of non zero homogeneous degree $d$ polynomials in three variables up to scaling, where
$\delta_d:= \frac{d(d+3)}{2} $. Given a point $p \i …
- 3,245
3
votes
1
answer
394
views
Does passing through a point in general position cut down the dimension by one?
Let $\mathcal{D} \approx P^{\delta_d}$ be the space of homogeneous degree $d$
polynomials in three variables (up to scaling), where $\delta_d = \frac{d(d+3)}{2}$. A point
$p\in \mathbb{P}^2$ gives u …
- 3,245
3
votes
0
answers
240
views
What is known about analogous results of Kazdan and Warner in higher dimensions?
First let me state a Theorem due to Kazdan and Warner:
``Let M be a compact two dimensional orientable manifold. Let
$f: M \rightarrow \mathbb{R}$ be a function that has the same
sign as $\chi(M)$ …
- 3,245
3
votes
0
answers
283
views
Do J-holomorphic curves "very nearly" fail to be an immersion near the bubbling points?
Let $u_{t}: \mathbb{P}^1 \longrightarrow \mathbb{P}^2$ be a family
of degree $2$ maps defined (for $t$ small and non zero) by
$$u_t([X,Y]) := [X^2, t Y^2, XY].$$
Note that as $t$ goes to zero, $u_t …
- 3,245
3
votes
2
answers
808
views
Can you ``perturb'' a submanifold to intersect transversally with any other smooth submanifo...
Let $\mathcal{D} \approx \mathbb{P}^{\delta_d}$ be the space of homogeneous
degree $d$ polynomials in three vriables, where
$\delta_d = \frac{d(d+3)}{2}$. Let
$$ X \subset \mathcal{D} \times \mathb …
- 3,245