# Questions tagged [smooth-manifolds]

Smooth manifolds and smooth functions between them. For manifolds with additional structure, see more specific tags, such as [riemannian-geometry]. For more topological aspects, see [differential-topology].

**6**

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**1**answer

227 views

### Volume comparison on Grassmannian

Let $G(r,n-r)$ be the Grassmannian, which can be identified with the space of all rank $r$ projection matrices in $\mathbb{R}^n$. Let $\mu$ be the uniform measure on $G(r,n-r)$. For any $\lambda_1,......

**3**

votes

**0**answers

84 views

### Volume ratio of balls in Grassmannian with different metric

Let $G(r,n-r)$ be the Grassmannian, which can be identified with the space of all rank $r$ projection matrices in $\mathbb{R}^n$. Let $\mu$ be the uniform measure on $G(r,n-r)$. For any $\lambda_1,......

**7**

votes

**0**answers

55 views

### Uniqueness of Fano varieties

It is a theorem of Kollár–Miyaoka–Mori that there is a finite number of deformation families of smooth, complex Fano $n$-folds for each $n$ (hence also a finite number of diffeomorphism types).
My ...

**-1**

votes

**0**answers

79 views

### (Un)orientable of smooth manifolds of $M^{d+1}$ and its boundary $N^d$ [closed]

Let $M^{d+1}$ be the ($d+1$)-dimensional smooth manifold with a boundary $N^d$ as a $d$-dimensional smooth manifold.
Are these statements true:
Statements 1-3:
If $M^{d+1}$ is orientable, then $N^d$...

**2**

votes

**1**answer

102 views

### Are normal coordinates the same as Cartesian coordinates in flat space?

Let $\gamma_v$ be the unique maximal geodesic with initial conditions $\gamma_v(0)=p$ and $\gamma_v'(0)=v$ then the exponential map is defined by
$$\exp_p(v)=\gamma_v(1)$$
If we pick any orthonormal ...

**15**

votes

**1**answer

350 views

### Wu formula for manifolds with boundary

The classical Wu formula claims that if $M$ is a smooth closed $n$-manifold with fundamental class $z\in H_n(M;\mathbb{Z}_2)$, then the total Stiefel-Whitney class $w(M)$ is equal to $Sq(v)$, where $v=...

**1**

vote

**0**answers

133 views

### Extending Green's theorem from very special regions to more general regions

Green's theorem
Let $C$ be a positively oriented and consists of a finite union of disjoint,piecewise smooth simple closed curve in a plane, and let $D$ be the region bounded by $C$. If $P$ and $Q$ ...

**0**

votes

**1**answer

79 views

### Same fiber of induced covering map [closed]

Consider a holomorphic map $h: X \to E$ between compact, connected, complex analytic manifolds Let $p: \tilde{E}\to E$ be the universal cover, and denote by $\tilde{h}: \tilde{X}\to\tilde{E}$ the pull-...

**0**

votes

**0**answers

54 views

### The idealizer of the space of vector fields with vanishing divergence

The space of smooth vector fields on a manifold is counted as a Lie algebra with the usual Lie bracket structure.
Is there a Riemannian manifold of dimension at least $2$ which satisfies either of ...

**6**

votes

**1**answer

196 views

### Milnor immersion of circle, disks, and a ball

Milnor surprisingly found an immersion of a circle in the plane which bounds two "incompatible" immersed disks (see [1] for example, picture included). I think I can glue these two disks together to ...

**4**

votes

**0**answers

121 views

### Integration over a Stiefel manifold

Let $V_{m}(\mathbb{R}^n)$ be the Stiefel manifold, which is the space of all $n \times m$ matrices such that the $m$ columns are orthonormal.
I want to calculate the volume of the following subset of ...

**0**

votes

**0**answers

30 views

### Integrating over some domain of the Stiefel manifold to analyze its support

Define the square $d$ dimensional Stiefel manifold as
$$V_{d} = \{ R \in \mathbb{R}^{d \times d} : R ^\top R = I_d \} .$$
How does one integrate on this manifold over a domain defined as $\{ R \in V_{...

**5**

votes

**2**answers

220 views

### Existence of an isotopy in Riemannian manifold

Let $(M,g)$ be a Riemannian manifold, and $p,q\in M$ be two fixed points. We assume $p,q$ are close enough. Say, we assume $p$ and $q$ are in the same normal coordinate chart. It is clear that there ...

**4**

votes

**0**answers

130 views

### Finite good covers on smooth manifolds

Let $M$ be a connected smooth manifold that is not necessarily compact but has the homotopy type of a finite CW complex.
Does $M$ admit a finite good cover? (i.e. a finite cover by contractible ...

**1**

vote

**0**answers

64 views

### Infinitesimal matrix rotation towards orthogonality

TLDR; I am trying to prove the existence of an infinitesimal rotation which always moves a matrix "closer" to being orthogonal.
Setting
In this setting, we have a matrix $W \in \mathbb{R}^{n \times ...