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Results tagged with differential-topology
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user 90655
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”.
18
votes
1
answer
908
views
Consequences of Gromov's Conjecture
In Peter Petersen words, Gromov Betti number estimate is considered one of the deepest and most beautiful results in Riemannian geometry; which asserts that
Theorem (Gromov 1981). There is a constant …
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4
votes
What are the important geometric-topological consequences of 4-dimensional version of Gauss-...
A good paper in this direction is "Some implications of the generalized Gauss-Bonnet theorem" written by Bishop and Goldberg which proved the following two theorems
Theorem 1.1. A compact and o …
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0
votes
1
answer
145
views
Is Kakutani's Theorem true for the Euclidean plane? [closed]
The following analogue of Kakutani's theorem has been proved by F. J. Dyson (MR44620)
Theorem (F. J. Dyson 1951): Let $\Bbb S^2$ be the surface of a sphere with center $Z$ in Euclidean $3$-space …
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3
votes
2
answers
408
views
A question on continuous maps from Möbius to itself
Let $M$ denotes the Möbius strip. Then is it true that
For every continuous map $f:M\to M$ there is $x\in M^\circ$ ($x\notin\partial M$) such that $f(f(x))=x$?
- 4,195
-3
votes
1
answer
374
views
Is the 2-dimensional Gauss-Bonnet theorem applicable in higher dimensions? [closed]
This is a cross-post of this MSE post that users commented that it is appropriate for MO.
I want to know
Question: Is the 2-dimensional Gauss-Bonnet theorem applicable (any topological or geometrica …
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3
votes
0
answers
238
views
About Riemann curvature tensor of local reflection
Let $\alpha: [a,b]\to M$ be an embedded curve in a Riemannian manifold $(M,g)$ and let
$p$ be a point in $M$, not on the curve $\alpha$. If $p$ is close enough to $\alpha$, there exists a
unique geode …
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14
votes
2
answers
1k
views
What are the important geometric-topological consequences of 4-dimensional version of Gauss-...
The Gauss–Bonnet theorem in differential geometry is an important statement about surfaces which connects their geometry to their topology and has very important applications to Riemann surface theory …
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3
votes
1
answer
181
views
Open neighbourhood of a point of space of Riemannian metrics
Let $M$ be a finite-dimensional compact smooth manifold and
$$\mathcal{M}et(M) = \{ g : g \text{ is a Riemannian metric on }M \}.$$
Q1-a: What metrics $g$ are very close to the given metric $g_0$? I.e …
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3
votes
0
answers
309
views
Correction to Milnor's h-cobordism book
This is a cross-post from MSE.
These four screenshots from milnor's book baffled me a bit (pages 24, 50, 51 and i-iii resp.):
In first one, there is no Theorem 3.1 in the book, but there i …
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2
votes
1
answer
652
views
Why non closed differential forms do not play important role for the topology of a manifold?
Cross-posted from MSE.
I know that De Rham cohomology reveal some properties of the topology of smooth manifolds by finding closed differential $k$-forms $\mathsf{d}\omega=0$ that are not exact $\ome …
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0
votes
1
answer
532
views
Is the meaning of "irreducible manifold", "not reducible to other manifold"?
This is a cross post of MSE.
Q1: What does "irreducible manifold" mean (not definition)?
My understanding of "irreducible manifold" is "is not reducible (homotopic or deformation or homeomorph or be …
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6
votes
1
answer
227
views
Does $\pi_k(M)\neq 0$ implies $\operatorname{ind}(\gamma) < k$?
Cross post from MSE. and sorry if this is an obvious question.
Here is a line of proof of Theorem 1.15 from
Brendle, Simon, Ricci flow and the sphere theorem, Graduate Studies in Mathematics 111. Prov …
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4
votes
1
answer
208
views
Existence non-trivial parallel $p$-form implies non-triviality of $p$-th cohomology group us...
Cross-post from MSE.
Suppose $(M,g)$ be a closed Riemannian manifold. Because every parallel (nontrivial) $p$-form $\omega$ is harmonic so the $p$-th Betti number should be positive i.e. $b_p\geq 1$. …
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8
votes
0
answers
405
views
What specifically is the gap in Aubin's argument about positive Ricci curvature that Paul Eh...
In his paper [2], Paul Ehrlich write
In [1], Aubin stated a theorem which implied as a corollary that if a manifold
$M$ admits a Riemannian metric with nonnegative Ricci curvature and
all Ricci curva …
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1
vote
1
answer
239
views
Reference for non-parallel harmonic $k$-forms
I want to get some deep understanding on closed orientable Riemannian manifolds admitting $k$-forms ($k\geq 2$) $\omega$ that satisfices the following conditions:
$$\nabla \omega\neq 0,\quad \Delta\om …
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