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Tagged Questions

4
votes
0answers
135 views

Embedding tower in low codimension

If $F$ is a suitably nice functor from manifolds to spaces, it has a degree $k$ "polynomial" approximation $T_k F$ in the sense of embedding calculus. We set $T_\infty F := \mathrm …
13
votes
3answers
404 views

Does a *topological* manifold have an exhaustion by compact submanifolds with boundary?

If $M$ is a connected smooth manifold, then it is easy to show that there is a sequence of connected compact smooth submanifolds with boundary $M_1\subseteq M_2\subseteq\cdots$ suc …
2
votes
1answer
330 views

About Sectional Curvature

In a paper by Yann Ollivier: Let $x$ be a point in $X$, $v$ a small tangent vector at $x$, $y$ the endpoint of $v$, $w_x$ a small tangent vector at $x$, and $w_y$ the parallel tra …
7
votes
4answers
432 views

geometric interpretation of Lie bracket

On page 159 of "A Comprehensive Introduction To Differential Geometry Vol.1" by Spivak has written: We thus see that the bracket $[X,Y]$ measures, in some sense, the extent to whi …
5
votes
1answer
327 views

Cancellation law for $M^n\times \mathbb R= N^n\times \mathbb R$.

Assume $M^n$ and $N^n$ are null bordant, i.e. each can be realized as boundary of an $n+1$ dimensional manifold. Suppose $M^n \times \mathbb R$ is homeomorphic to $N^n\times \mathb …
12
votes
2answers
385 views

Are there results from gauge theory known or conjectured to distinguish smooth from PL manifolds?

My question begins with a caveat: I sometimes spend time with topologists, but do not consider myself to be one. In particular, my apologies for any errors in what I say below — c …
0
votes
1answer
133 views

Does some type of curvature require the space be an embedded manifold in a higher-dimensional space? [closed]

Assuming an appropriate definition of 'curvature', is there a theorem that says: "At least n+1 dimensions are necessary for a particular curvature to exist in an n-dimensional spac …
19
votes
3answers
639 views

Connected sum of topological manifolds

A definition of the connected sum of two $n$-manifolds $M$ and $M'$ begins by considering two $n$-balls $B$ in $M$, $B'$ in $M'$, and glueing the varieties $M\setminus \mathring B …
8
votes
3answers
300 views

Characterizing Hessians among symmetric bilinear tensors

I apologize in advance if this is somewhat elementary, but: Let $(M,g)$ be a compact Riemannian manifold. Is there a "characterization" of which symmetric bilinear tensors $B\i …
0
votes
0answers
165 views

What is topological surgery conjecture?

I'm a novice in the surgery theory, and I'm encountering the word "(4-dimensional) topological surgery conjecture", which I can't not find the definition. Could anyone help me?
0
votes
0answers
118 views

A question from Hamilton’s Ricci Flow book by bennett chow

On page 3 of the book before exercise 1.2, is written: "torsion free is a compatibility condition with the differentiable structure". I correctly do not understand how torsion-free …
3
votes
1answer
80 views

A k-form is thought of as measuring the flux through an infinitesimal k-parallelepiped

On the wikipedia has written "A $k$-form is thought of as measuring the flux through an infinitesimal $k$-parallelepiped." How does a $k$-form do this? if this sentence is right, t …
8
votes
2answers
789 views

homotopy type of embeddings versus diffeomorphisms

Previously, I asked a question on mathoverflow comparing smooth embeddings and diffeomorphisms, which received a very interesting and somewhat unexpected answer by Agol. I now ask …
1
vote
1answer
188 views

special Lagrangian n-Torus has Tubular neighbourhood?

Let $\imath :T^{n}\rightarrow X$ is a special Lagrangian n-Torus so that $\imath(T^{n})=L$ and all small special Lagrangian deformations of $L$ are flat then why $L$ has Tubular ne …
3
votes
2answers
177 views

When a Riemannian manifold is of Hessian Typ

When a Riemannian manifold is of Hessian Type (i.e., a Riemannian manifold which its metric is Hessian)

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