# Questions tagged [dg.differential-geometry]

Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.

7,791
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### Concrete descriptions of $S^1$-bundles over smooth manifold $Y$ underying a K3 surface

Let $Y$ be the smooth manifold underlying a K3 surface. As a manifold, $Y$ is diffeomorphic to $\{[x_0:x_1:x_2:x_3]\in\mathbb{C}P^3\colon X_0^4+x_1^4+X_2^4+X_3^4=1\}$. It is well known that $H^2(Y,\...

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### $L^2$ norm of scalar curvature

As mentioned by Wilie Wong, I modified to the following verison:
Let $M$ be a closed smooth $4$ manifold.
Q
Suppose that $c>0$ is any positive number, can we find a Riemannian metric $g$ on $M$, ...

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43
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### Is there a general definition of twisted Real equivariant cohomology theory?

There are some classical examples of Real equivariant cohomology theories and twisted cohomology theories, including equivariant KR-theory in Atiyah and Segal's paper, and the more general ...

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### Expressing a vector valued function in terms of its derivatives

Consider a function
$$
f:\mathbb{R}^n\rightarrow\mathbb{R}^m
$$
given by $m$ functions $f_i:\mathbb{R}^n\rightarrow \mathbb{R}$ that we can assume to be polynomials in $x_1,\dots,x_n$.
Does there ...

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86
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### Construction of holomorphic line bundles on complex torus

This is an argument for constructing positive line bundles on complex torus. From some knowledge of Abelian varieties, such as Riemann conditions, we know that it is wrong. But I don't know where this ...

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40
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### Example on pseudo-groups

Definition:
A pseudo-group is a collection $\mathcal{G}$ of (locally defined) invertible smooth diffeomorphisms of a manifold $M$. The simplest example of a pseudo-group is the collection of all ...

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2
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369
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### The convex hull of a manifold whose cobordism class is trivial

Let $M$ be a compact orientable $n$ dimensional manifold. Assume that $M$ has trivial cobordism class.
Is there an embedding of $M$ in some Euclidean space $\mathbb{R}^m$ such that the convex ...

2
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1
answer

192
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### Lie algebroid in algebraic geometry

When I did net-surfing at home, I met some geometric backgrounds of Lie algebras and encountered the concept of Lie algebroids. In differential geometry, a Lie algebroid seems to be defined as ...

5
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+100

### Is the volume functional analytic in the space of embeddings? What about locally?

Let $(M^{n+1},g)$ be an analytic Riemannian manifold and let $\Sigma^n$ be a closed analytic manifold. Denote by $\operatorname{Emb}(\Sigma, M)$ the space of all smooth (or maybe analytic) two-sided ...

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1
answer

47
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### Cross product of two infinitesimal bendings

Let $M$ be a smooth (embedded or immersed) surface in $\mathbb{R}^3$.
Let $Z_1,Z_2$ be two vector fields along $M$, thought of as $\mathbb{R}^3$-valued functions, satisfying the following differential ...

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### Spectral theory of infinite volume hyperbolic manifolds

I have a question about the discrete spectrum of the Laplace operator on hyperbolic manifolds with infinite volume. I understand the case of infinite area surfaces: see Chapter 7 (Sections 1 and 2) of ...

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### Curves traced out by the centers of mass of rolling convex shapes

Question:
which kind of curves can be traced out by the center of mass of a rigid compact convex shape of uniform density that rolls along the x-axis without slip?
Formulatd differently: are there ...

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34
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### A map in cohomology for a pair of foliations

Let $M$ be a compact manifold with a pair of transverse foliations $\mathcal{F}$ and $\mathcal{G}$, i.e.
$$
TM=T\mathcal{F}\oplus T\mathcal{G}.
$$
Restriction of differential forms to the leaves of $\...

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202
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### Proof of Ehresmann's theorem

In Huybrechts' book Complex geometry: An introduction p.269, Proposition 6.2.2, the author gives a proof of the following theorem
(Ehresmann)
Let $\pi:\mathcal X\to B$ be a proper family of ...

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99
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### 2-shifted 2-form on the classifying stack 𝐵𝐺

Let $G$ be a reductive group. A $2$-shifted $2$-form on the classifying stack $BG$ is by definition a a morphism of quasi-coherent complexes
\begin{equation}
\mathcal O_{BG}\rightarrow (\wedge^2 \...

6
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1
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392
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### Derivations on the continuous functions of a manifold

For a manifold $M$ a vector field is a derivation of the algebra $C^{\infty}(M)$ of smooth functions on $M$. What happens if look instead as derivations on the continuous functions of a manifold. I ...

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86
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### Question about deformation of the metirc on a Riemannian manifold

I'm a bit confused with the deformation of the metric on a given Riemannian manifold $(M,g)$ with a smooth boundary. How can we deform the metric $g$ such that it is a product near $\partial M$, ...

7
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1
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232
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### On fixed point sets of actions of compact Lie groups

Let a compact Lie group $G$ act smoothly on a compact smooth manifold $M$. For any compact subgroup $H\subset G$ denote by $E^H$ the image in $M/G$ of the fixed point set of $H$ in $M$.
Is it true ...

3
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88
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### Geometric intuition behind definition of $\delta$-necklike points of the Ricci flow

In "The Ricci Flow: An Introduction", the authors define a $\delta$-necklike point of the Ricci flow as a point $(x, t)$ where $$\|\text{Rm} - R (\theta \otimes \theta)\| \leq \delta \|\text{...

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1
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50
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### Changing the system of PDE by diffeomorphism and differentiate a composition

This problem comes from the book Hamilton's Ricci flow.
Given a smooth functional $f$, and following system. $$\partial_t f=-(\Delta f+R)$$ If there exist a 1 parameter family of diffeomorphism $\Psi(...

2
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66
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### Norm of a $(1, 1)$ form on a Kähler manifold

Given a Kähler manifold $(M, g)$ what is the convention for defining the inner product on two $(1,1)$ forms $\alpha = \sqrt{-1}\alpha_{i \bar k} dz_{i} \wedge d \bar{z}_{k}$ and $\beta = \sqrt{-1}\...

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### Prescribing variations that preserve the area

Let $(M^3,g)$ be a Riemannian manifold and let $\varphi : \Sigma \to M$ be a two-sided embedding of a closed surface into $M$, with a unit normal $N$. Suppose that $\varphi$ is a regular point of the ...

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92
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### $C^0$ norm is bounded by $L^{14}$ norm

Let $M$ be a closed manifold of dimension $6$, and we look at the collection of smooth functions on $M$ which satisfy:
\begin{align*}
||f||_{C^0}\leq C\big(||f||_{L^{14}}^2+1\big)
\end{align*}
for ...

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votes

1
answer

116
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### Pull back a vector field [closed]

In Voisin's book Hodge theory and complex algebraic geometry, I Section 9.1.2, p.223, the author writes:
Let $\phi:\mathcal X\to B$ be a family fo complex manifolds. The differential $\phi_*$ is a ...

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0
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24
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### Results on compact slices in a regular foliation

Let $(M,\mathcal{F}$) be a smooth and regular foliation (not necessarily of comdimension 1). I am wondering if there are known (partial) results on the existence of compact, connected submanifolds $F\...

2
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### Connection between a function and its usage in geometry [closed]

I know nothing about geometry, but I found a function which seems to have something to do with geometry.
This function is, $$f(x,y,z) = \dfrac{(x,y,z)}{\sqrt{1 + x^2 + y^2 + z^2}}$$
where $x,y,z$ is ...

3
votes

1
answer

107
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### Convex sphere in R^3

Is every convex sphere (in the sense of Alexandorff, which is the boundary of some convex body in $\mathbb{R}^3$) with Alexandorff curvature $\geq 1$, bi-Lipschitz to the unit round sphere in $\mathbb{...

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44
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### composition of two tangent vector fields and connection [closed]

Let $X, Y$ be two tangent vector fields on a manifold.
Consider their composition:
$(X \circ Y)(f) = X(Y(f)) = X^i\frac{\partial Y(f)}{\partial u^i} = X^i\frac{\partial Y^j}{\partial u^i}\frac{\...

2
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43
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### Asymptotic expansion of the Hessian of the distance function

This question originates from another question. Big thanks to MySheperd whose answer to that question clarified my thoughts.
Let $(M,g)$ be an $n$-dimensional complete Riemannian manifold and $r$ is ...

3
votes

1
answer

109
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### Area of a deformation of a closed surface

Let $(M^3,g)$ be a complete Riemannian manifold. Fix a two-sided immersion $\varphi : \Sigma^2 \to M$ from a closed surface into $M$, with unit normal $N$. Given $f \in C^\infty(\Sigma)$ and $\alpha : ...

11
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1
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279
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### How wild can an open topological 3-manifold be if it has a compact quotient?

Let $M$ be an open, simply connected, 3-manifold. Suppose $M$ admits a properly discontinuous, co-compact topological action by a finitely generated group.
Question 1: If $M$ is 1-ended, must it be ...

4
votes

1
answer

104
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### Question about differential operators in a completely non-integrable distribution

Say I have two integrable codistributions
$$ U = \langle du^1, \ldots, du^m \rangle, \qquad Z = \langle dz^1, \ldots, dz^N \rangle $$
on a manifold $M$, with $N >> m$. Suppose that the ...

4
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0
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96
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### Convex core and geometric finiteness of negatively curved manifolds

I am reading a paper on hyperbolic geometry where they are using the concept of "convex core" in the context of "geometric finiteness". Roughly, this means (from Definition F4 of a ...

1
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1
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73
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### Pull-back of factor of automorphy

Let $M=\mathbb C^g/ \Gamma$ be a complex tori and $E$ a be a holomorphic vector bundle of rank $r$ over $M$. Then $E$ is characterised by factor of automorphy, i.e. a holomorphic map $J:\Gamma\times\...

2
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### Why does the solution to pendulum problem with the geometric approach of Jacobi metric does not correspond to the solution with Lagrangian approach? [closed]

When we solve the pendulum problem with EL equation, we get to the differential equation $\ddot{q}+\frac{g}{l}\sin q=0$
but when I apply the substitution $t \rightarrow t\sqrt\frac{g}{l}$ and ...

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0
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43
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### Critical points of function-curvature

As a side effect of the COVID-19 pandemic exponential growth became a buzz word that was "copy-pasted" a lot in public discussion.
It may be assumed that the general public can't make sense ...

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0
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93
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### Ends of a negatively curved Riemannian manifold

Let $M$ be a complete Riemannian manifold. Let us use the standard definition of "end", for example, as in this article. If $M$ has non-negative Ricci curvature, it is well-known that it has ...

7
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1
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166
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### Harmonic functions on complete Riemannian manifolds

I have started reading a paper of Colding and Minicozzi, where they prove that on a complete Riemannian manifold $M$ of non-negative Ricci curvature, the space of harmonic functions of growth order at ...

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0
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61
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### Non-Kähler Hermitian homogeneous spaces

I am looking for examples of compact homogeneous space endowed with the structure of a non-Kähler Hermitian manifold.

9
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1
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375
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### Examples of 6-manifolds without an almost complex structure

Question: I am searching for examples for closed (hence orientable ), smooth $6$-manifolds without an almost complex structure.
Finding such an example is equivelant to finding a manifold where the ...

1
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1
answer

218
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### A problem arising from reading a lecture on the Yamabe problem of how the Hölder inequality is used

I'm reading Tawfik - The Yamabe problem: the PDE is
$$
\Delta \varphi+h(x) \varphi=\lambda f(x) \varphi^{q-1}. \label{1}\tag{1}
$$
Theorem (Yamabe). For $2<q<N=N=2 n /(n-2)$, there exists a $C^{\...

2
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1
answer

135
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### Decomposition of tensors

It is well known that every traceless, symmetric $2$-tensor can be decomposed uniquely into a Lie derivative part and a Codazzi part. Is there an analog for totally symmetric $k$-tensors?

1
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1
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149
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### Can one explore a surface along ‘piecewise planar’ curves?

Suppose $d\in \{3,4,\dotsc\}$ and $A\subseteq \mathbb{R}^d$ is non-empty, open and connected with its complement $A^c$ connected too and $\text{int}(A^c)\neq \emptyset$. Its boundary $S:=\partial A$ ...

3
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### Intersection number for 4 manifold with boundary

Let $X$ be a closed oriented smooth $4$-manifold. Suppose there is an embedding $\Sigma\to X$, it is known that the self-intersection number satisfies $[\Sigma]\cdot [\Sigma]=\pm\int_\Sigma c_1(N)$, ...

3
votes

1
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299
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### Hessian of the distance function--comparison with the space form with constant sectional curvature 0

Let $M$ be an $n$-dimensional complete Riemannian manifold and $r$ is the distance function to a fixed point.
The Hessian comparison theorem says that if the sectional curvature of $M$ is bounded (...

2
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0
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53
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### Question about Clifford volume element

I'm a little confused with the following:
Let $M$ be a $m$ dimensional Riemannian manifold and $e_1,\cdots,e_m$ be a local orthonormal base of $TM$. Let
$$
\omega_\mathbb{R}=c(e_1)\cdots c(e_m)
$$
...

3
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0
answers

45
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### Decomposition about splitting of symmetric spaces of compact type

I get stuck in the following question:
Why does a locally symmetric space of compact type $M$ split locally irreducible components of dimension $\geq 2$ which are Einstein? In particular, why are all ...

2
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81
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### Questions about symmetric spaces

I'm a little confused with the following questions:
(1) Why does a symmetric space $M=G/K$ of compact type have $\mathcal{R}^M\geq 0$?
(2) Moreover, why does $\chi(M)\neq 0$ if and only if ${\rm rk}\ ...

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132
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### Why are we interested in spectral gaps for Laplacian operators

Let $M$ be a Riemannian manifold and let $\Delta$ be its Laplacian operator. There is a large literature on a spectral gap for such a $\Delta$, that is, finding an interval $(0,c)$ which does not ...

6
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1
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215
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### Can every smooth space curve be realized as an origami curved crease?

Many years ago, Ron Resch told me that he proved that every smooth simple space curve $C$ could be realized as a curved crease $\gamma$ in the interior of a piece of paper.
He never published this (as ...