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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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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 …
- 4,195
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 …
- 4,195
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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4
votes
0
answers
418
views
What are your common strategies/remedies when your new theory/idea stuck in most cases?
Sorry if this is not a suitable post for MO.
Sometimes after reading the origin of a theory/idea in differential topology I put myself in the shoes of that mathematician and ask myself, Did you do the …
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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 …
- 4,195
4
votes
1
answer
309
views
Existence parallel vector fields and its effect on the topology of manifolds (Karp's Thesis)
It seems that there is no digital copy of Leon Karp's Ph.D. thesis
L. Karp, Vector fields on manifolds, Thesis, New York Univ., 1976.
on internet and his paper excerpted from his thesis is very brief …
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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 $f(f(x))=x$?
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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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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
578
views
Why trace is more natural than (preferred to) determinant for smooth map $f:M\to N$?
Cross-post from MSE.
For a continuous map $f:(M,g)\to (N,h)$, between Riemannian manifolds $(M,g)$ and $(N,h)$ we can pullback $h$ by $f$. Most experts take the trace from this new tensor and work wit …
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2
votes
Is the meaning of "irreducible manifold", "not reducible to other manifold"?
Summary of comments and other sources
There are at least 4 similar concepts:
Irreducible smooth manifold: As Ryan Budney said, "Regarding high dimensions, generally irreducible manifolds do not exist …
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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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