# Questions tagged [manifolds]

A manifold is a topological space that locally resembles Euclidean space near each point. More precisely, each point of an n-dimensional manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension n.

**1**

vote

**0**answers

17 views

### Matrix trace minimization of quadratic and linear terms under orthogonal manifold constraints

How would one solve the following orthogonal manifold problem?
$\max_{\{X : X^\top X = I\}} \text{tr}(X^\top A X - X^\top B)$ where $A \succeq 0$
I've seen one method that successively performs the ...

**0**

votes

**1**answer

327 views

### Metrics on derived smooth manifolds

Derived geometry explains how to remove the transversality condition and make sense out of a nontransversal intersection.
For example, if $X$ and $Y$ are embedded submanifolds of a manifold (or ...

**7**

votes

**1**answer

220 views

### Can a hyperbolic manifold be a product?

I was interested in whether a manifold which admits a metric of constant sectional curvature can be homotopy equivalent to a product of non-contractible manifolds. Of course, there are three cases: ...

**9**

votes

**0**answers

255 views

### History of the definition of smooth manifold with boundary

I am trying to determine the earliest source for the definition of smooth ($C^\infty$) manifold with boundary. Milnor and Stasheff (1958) give a definition, but a scrutiny of that definition shows it ...

**2**

votes

**0**answers

161 views

### Triple link in a 5-sphere — Proposal

In this post I would like to propose a triple link in a 5-sphere.
Let us start with the following gluing into a 5-sphere:
$$S^5=(D^2_{} \times T^3_{}) \cup_{T^4} ({S^5 \smallsetminus D^2 \times T^3})$...

**5**

votes

**1**answer

150 views

### harmonic coordinates on non-compact manifolds

Is it possible to show the existence of harmonic coordinates (e.g., on uniform-sized balls) on certain classes of non-compact Riemannian manifolds? For example, one may expect that such harmonic ...

**1**

vote

**2**answers

109 views

### Self-adjoint extensions for pseudo-differential operators

The class $\Sigma^1$ of symbols on $\mathbb R^{2n}$ is made with $C^\infty$ functions $a$ of $X=(x,\xi)\in \mathbb R^n\times\mathbb R^n$ such that
$$
\vert\partial_X^\alpha a\vert\le C_\alpha(1+\vert ...

**2**

votes

**0**answers

58 views

### Vector bundle endomorphism diffeomorphism invariant?

Let's say we have a vector bundle $V$ on a compact Riemannian manifold $M$ (of dimension $m$, with metric $g$). Given a differential operator $P$, acting on the sections of $V$, of the form: $$P = \...

**8**

votes

**0**answers

139 views

### $k$ times differentiable but not $C^k$ manifold

I asked the following question on Math Stack Exchange 3 months ago but got no answer. So maybe Math Overflow is a more suitable place for such a question:
I cannot find the notion of $k$ times ...

**1**

vote

**0**answers

79 views

### Manifold with no closed components?

Let $M$ be a manifold with boundary. Reading some papers on $3$-manifolds I have come across some statements where they require that: ”$M$ has no closed components.”
What does this mean? The ...

**9**

votes

**0**answers

216 views

### Is it possible to glue together complex manifolds?

In the case of Riemannian manifolds, there are ways to take two manifolds and glue them together to get a new Riemannian manifold. For example, taking connected sums in local regions where the two ...

**16**

votes

**1**answer

723 views

### Can one determine the dimension of a manifold given its 1-skeleton?

This may be an easy question, but I can't think of the answer at hand.
Suppose that I have a triangulated $n$-manifold $M$ (satisfying any set of conditions that you feel like). Suppose that I give ...

**0**

votes

**0**answers

28 views

### dual and intersection of a simplex

In a triangulation $\Gamma$ of a (oriented) 2-manifold, consider a 2-simplex labeled by ($123$), where $1,2,3$ denote the order of vertices. Consider the dual $\Gamma^*$ of $\Gamma$, and then denote ...

**2**

votes

**0**answers

153 views

### Reference for a proof of a Theorem by Joseph Wolf

We know that Lie Groups are parallelizable, I was looking for a version of the converse and came across this:
https://books.google.com/books?id=w4bhBwAAQBAJ&pg=PA115 in Introduction to Smooth ...

**8**

votes

**1**answer

385 views

### Critical dimensions D for “smooth manifolds iff triangulable manifolds”

I am aware that at least for lower dimensions,
"smooth manifolds iff triangulable manifolds"
at least for dimensions below a certain critical dimensions D.
My question is that for
...