# Questions tagged [dg.differential-geometry]

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

8,691
questions

1
vote

1
answer

86
views

### Lower bound on injectivity radius at one point implies lower bound on injectivity radius for a closed manifold

I’m interested in a closed Riemannian manifold $(M^n,g)$ with $sec<0$ and $diam(M)\leq D$. My question is:
If at some point $p\in M^n$, the injectivity radius $injrad(p)\geq1$, then can we get $...

2
votes

0
answers

57
views

### First examples of Lie-Rinehart algebras that are not coming from Lie algebroids

I heard the idea of a Lie-Rinehart algebra first time from an algebraist.
I noticed there is a similarity between description of Lie algebroid on a manifold and the algebraic notion of Lie-Rinehart ...

3
votes

1
answer

260
views

### Zeros of a function defined on $\mathbb{S}^2 \times \mathbb{S}^2$

Let $u$ be a smooth function on the sphere, and for each $y \in \mathbb{S}^2$, let $R_y$ be the $180^\circ$ rotation about the vector $y$. For each pair $(x, y) \in \mathbb{S}^2 \times \mathbb{S}^2$, ...

0
votes

0
answers

92
views

### $C^\infty$-coring

We know that there the so called smooth algebras also known as $C^\infty$-rings. They can play an important role in modern treatment of differential geometry. Is there a coring analogue?

2
votes

0
answers

31
views

### De Rham product decomposition theorem in a particular setting

Let $G$ be a Lie group, and $H$ a Lie subgroup of $G$ such that $G/H\sim \mathcal M$ is a homogeneous space diffeomorphic to $\mathbb{R}^n$, equipped with an invariant Riemannian metric $g$. If this ...

3
votes

0
answers

41
views

### Applying Li-Yau-Hamilton estimate to heat kernel

In Li-Yau gradient estimate one apply maximum principle to the quantity $F(x,t)=t(|\nabla u|^2-\alpha u_t)$ where $u$ is a positive solution of heat equation on $M$. It seems that maximum principle ...

0
votes

0
answers

64
views

### Uniqueness of antipodal points on compact manifolds

Suppose $(M,g)$ is a compact smooth Riemannian manifold without boundary. We say that two points $p$ and $q$ are antipodal if the distance between them is maximal distance in the manifold, that is to ...

2
votes

0
answers

45
views

### Terminology: generalized Laplacian of arbitrary signature

Let $(M,g)$ be a Riemannian manifold and $E$ any real or complex vector bundle. A linear partial differential operator $D:\Gamma(E)\to\Gamma(E)$ is called generalized Laplace operator, if its ...

1
vote

2
answers

92
views

### Question about the index of two elliptic operators over a 4-dimensional Riemannian manifold

I've already asked this question in: https://math.stackexchange.com/questions/4899825/question-about-the-index-of-two-elliptic-operators-over-a-4-dimensional-riemanni, and I've been suggested to ask ...

-1
votes

0
answers

48
views

### Relationship between a differential form and the tangent plane [closed]

Consider a point $(1, 2, 3) \in \Bbb R^3$, if we consider an exterior form of degree $1$, given by
$$w=\sum_{i=1}^3 a_{i} dx_{i}$$
So, for our case
$$
w(1, 2, 3) = \sum_{i=1}^3 a_{i}(1, 2, 3) \, dx_{i(...

1
vote

0
answers

117
views

### Conceptual understanding of the definition for Hermite-Einstein metrics

I'm studying holomorphic vector bundles $(E,h)$ on Kähler manifolds that admit a Hermite-Einstein metrics. Particularly, I'm trying to find the motivation for the definition.
An hermitian structure $...

5
votes

0
answers

82
views

+200

### Is there a canonical smooth structure on tame Fréchet orbit type stratifications?

In finite dimension orbit type stratifications, it is known that the orbit space $M/G$ resulting from an action of a proper Lie Group $G$ on a smooth manifold $M$, satisfying a set of certain ...

2
votes

1
answer

88
views

### Finite group extensions of lattices

I'm currently reading the proof of Geroch's conjecture in Lawson-Michelsohn's Spin Geometry book and in the proof of Proposition IV.5.8 that every Ricci-flat enlargeable manifold is flat the following ...

0
votes

0
answers

50
views

### Does convexity of boundary implies geodesic convexity?

I came across the following result (mentioned on Pg. 3 of this talk) that states that
If $D$ is an open connected subset of a complete Riemannian manifold with smooth metric then $\partial D$ convex ...

9
votes

1
answer

390
views

### Are limits of compact leaves compact?

Let $M$ be a compact smooth manifold, and $\mathcal{F}$ be a foliation on $M$. Assume that $L$ is a leaf of $\mathcal{F}$ for which there is $x\in L$ with the property that every neighborhood of $x$ ...

0
votes

0
answers

52
views

### Reference request: explicit formula for Lie derivative of matrix Lie groups

Let $M =End(n,\mathbb{C})$ be the space of complex matrices with adjoint action by $ U(n)$, i.e. acted by $B\rightarrow gBg^{-1}$ for $B\in End(n,\mathbb{C}),g\in U(n)$. Let $X_{\xi}$ be the vector ...

2
votes

0
answers

45
views

### Connection vs Exponential preserving maps

Connection Preserving Diffeomorphisms
The setting is a manifold $M$ equipped with a linear connection $\nabla$. Kobayashi & Nomizu [K&N §VI.1] define a connection preserving diffeomorphism (...

1
vote

0
answers

35
views

### Boundary behavior for submanifolds with bounded second fundamental form

I am interested in a boundary version of this question About hypersurfaces in R^n+1 with bounded 2nd fundamental form.
The question is as follows. Let $\Sigma^k\subset \Bbb R^n$ be a submanifold with ...

2
votes

2
answers

139
views

### Compute Christoffel symbols of sphere by embedding

In his answer V. Semeria, starts by taking
$$(y_1,\dots,y_{n+1})=\left(x_1,\dots,x_n,\sum_{i=1}^{n+1}x_i^2 -R^2\right)$$
Write $(\vec{e}_1,\dots,\vec{e}_{n+1})$ the canonical basis of $\mathbb{R}^{n+1}...

2
votes

1
answer

87
views

### Construction of Scherk's surface using soap films

I am currently interested in the differential geometry of minimal surfaces, and I have a rather trivial question regarding Scherk's surface (the one which can be parametrised by the real function $(x,...

2
votes

1
answer

101
views

### Existence of a spin map from a standard sphere to any closed Riemaninan manifold with nonnegative curvature operator

Let $S^m$ be a standard sphere of dimension $m=n+4k$, and let $M$ be any closed Riemaninan manifold of dimension $n$ with nonnegative curvature operator.
My question: Is there always a smooth spin map ...

1
vote

1
answer

141
views

### For proper group action on closed Riemannian manifold, must the union of orbits with non-unique closest points to a given point be of 0 volume measure

Let $(M,g)$ be a closed (compact without boundary) Riemannian manifold of finite dimension, with the volume measure $\mu:= \mu(E):=\int_{E}d\operatorname{vol}_g \forall E \in \mathcal{B}(M),$ the ...

1
vote

0
answers

81
views

### Question from Taubes' SW$\Rightarrow$ Gr

I am trying to understand Taubes' paper on SW$\Rightarrow$ Gr. I don't understand how either of the equations 2.16 or 2.17 appears, I would be happy to understand how the curvature term $F_a$ appears ...

3
votes

2
answers

299
views

### A paper of Borel (in German) on compact homogeneous Kähler manifolds

I am trying to understand the statement of Satz 1 in Über kompakte homogene Kählersche Mannigfaltigkeiten by Borel. Here is the statement in German
Satz I: Jede zusammenhängende kompakte homogene ...

2
votes

2
answers

444
views

### Are Chern classes always vertical?

Let $c_k \in H^{2k}(M, \mathbb{Z})$ be the $k$-th Chern class of the tangent bundle of a Hermitian manifold $M$.
Is $c_k$ necessarily vertical, i.e.
$$
c_k = \sum_{i_1,\dots, i_{k}} \alpha_{i_1 \dots ...

4
votes

0
answers

100
views

### A compact Kähler manifold that admits a homogeneous action of a non-reductive Lie group

Is it possible to have a compact Kähler manifold that admits a homogeneous action of a non-reductive Lie group? It seems not to be the case, but a precise argument of reference would be great!
Edit: ...

4
votes

1
answer

178
views

### Taubes' SW$\Rightarrow$ Gr

I am reading Taubes' paper on SW$\Rightarrow$ Gr and lost in some analysis, can anyone help me to see how to get equation 2.19 from equation 2.18? Is this some version of Kato for the Laplacian?

1
vote

0
answers

86
views

### A homogeneous manifold that does not admit an equivariant Riemannian metric?

Let $M = G/H$ be a homogeneous space, where $G$ is a Lie group and $H$ is a closed Lie subgroup. Can it happen that $M$ does not admit an invariant Riemannian metric?

1
vote

0
answers

57
views

### An integration formula that looks like polar coordinates in $\mathbb{R}^n$ [migrated]

Let $M$ be a complete $n$-dimensional Riemannian manifold with non-negative Ricci curvature. Let $x_0\in M$ and $\theta>1$ be fixed. Consider the function $f=\theta^{-1}d(\cdot, x_0)$, where $d$ is ...

1
vote

0
answers

76
views

### Differential operators and iterations of tangent bundle

Is there a relationship between higher order differential operators and higher tangent bundle viewed as bundle on the base manifold?

2
votes

1
answer

81
views

### Germs of left invariant differential operators on a group

Are there germs at the identity of linear differential operators on a group which are not germs at the identity of left invariant differential operators?
I feel like the answer is no but the statement ...

2
votes

1
answer

120
views

### Continuity of the volume function

Consider a continuous map $F:(a,b)\times\mathbb{S}^n\to\mathbb{R}^{n+1}$ such that for any $t\in(a,b)$, the map $F(t,\cdot)=F_t:\mathbb{S}^n\to\mathbb{R}^{n+1}$ is Lipschitz continuous. The $n$-...

3
votes

1
answer

142
views

### Analogue of vector for differential operators

A differential operators of order one is a vector field which is defined pointwise . Differential operators of order greater than one are not. The closest analogue to a vector is given by a germ of a ...

1
vote

0
answers

80
views

### Index and nullity of a short closed geodesic

Let $g$ be a reasonably smooth Riemannian metric on the n-dimensional sphere $S^n$. Call a closed geodesic $\gamma$ in $(S^n, g)$ short if, for every diffeomorphism $S^n \to S^n$, the image of at ...

2
votes

0
answers

76
views

### Simply connectedness of leaves of a foliation on an complex manifold

Now I'm searching about leaves of foliation in the following special setting.
Let $U,V$ be two holomorphic vector field on $\mathbb{C}^2$ s.t the Lie bracket $[U,V]=UV-VU=0$ and $U$ and $V$ spaned ...

2
votes

1
answer

156
views

### Identifying the circle bundle of the canonical line bundle $\mathcal{O}(-n-1)$ over a projective space $\mathbb{CP}^n$

It's not hard to see the following fact: the circle bundle of the tautological line bundle $\mathcal{O}(-1)\rightarrow \mathbb{CP}^n$ is $S^{2n+1}$, the unit sphere inside $C^{n+1}.$ I want to see ...

0
votes

1
answer

125
views

### Flow of a vector field

Consider a Riemannian manifold $(M^n , g)$ and let $d_p: M^n \to [0,\infty)$ be the distance function of $p \in M^n$. Then the flow lines generated by $\nabla d_p$ are radial geodesics from $p$. Also, ...

1
vote

0
answers

116
views

### What can we say when a module of differential is free?

Let $\mathbb{C}$ complex number.
$R=\mathbb{C}[x,y]/(f)(f\in \mathbb{C}[x,y])$
If the module of differential $\Omega_{R/\mathbb{C}}$ is free $R$ module of rank one,
what can we say about $R$.
How far ...

4
votes

0
answers

288
views

### Merits of derived geometry

What are the merits of derived geometry? More precisely, which specific mathematical problems that can be formulated without this machinery have been solved using it? If those problems exist, could ...

5
votes

2
answers

405
views

### Question about Neumann eigenvalues on manifolds

Question:
Let $\Omega\subset \mathbb{S}^2_+$ be any geodesically convex subset of the hemisphere $\mathbb{S}^2_+$. Then is it true that $\mu_1(\Omega)\geq \mu_1(\mathbb{S}^2_+)$ where $\mu_1$ is the ...

0
votes

0
answers

61
views

### About geodesic vector fields and the status of a classic problem on the number of closed geodesics

A classical problem in differential geometry is to determine whether every compact Riemannian manifold admits infinitely many geometrically distinct closed geodesics. A good reference on the subject ...

5
votes

1
answer

166
views

### Converging paths implies converging parallel transports along those paths?

Suppose we have a vector bundle $E$ with connection $\nabla$ over a smooth manifold $M$. Let’s also say we have a sequence of smooth paths $\gamma_n\in C^\infty([0,1],M)$ starting at the same point $\...

2
votes

1
answer

132
views

### Understanding the integral $\int_0^1\det(v(t),v'(t))dt$ where $v(t)$ is path in the plane

Let $v(t) : [0,1]\rightarrow\mathbb{C}^2$ be a smooth path, and let $v' := dv/dt$. I'd like to understand what the integral:
$$I(v) := \int_0^1 \det(v(t),v'(t))dt$$
tells us about $v$, where $\det(v(t)...

0
votes

0
answers

57
views

### Condition to show $\{ U \in \mathbb{R}^{n \times p}|\mathscr{A}(UU^{\top}) = b \}$ is (is not) a manifold

Consider $\mathscr{A}: S^{n\times n} \to \mathbb{R}^{m}$, $b \in \mathbb{R}^{m}$, I would like to know when $\mathscr{M}:=\{ U \in \mathbb{R}^{n \times p}|\mathscr{A}(UU^{\top}) = b \}$ is a manifold. ...

2
votes

1
answer

198
views

### Deriving the definition of vector bundle morphisms from Cartan geometry (a.k.a. why are they linear?)

I'm familiar with the definition of the category of vector bundles, but I'm trying to derive it from some first principles about general fiber bundles. My intuition is that vector bundles should be ...

3
votes

0
answers

218
views

### What is the image of a smooth map? [migrated]

Let $f: S^2 \to \mathbb{R}^n$ be a smooth map from the two-dimensional sphere to euclidean space. Let $X = \mathrm{Im}(f) \subset \mathbb{R}^n$ be the image topological space (note: the quotient ...

6
votes

0
answers

172
views

### What are compact manifolds such that GROWTH (of spheres volumes) is well approximated by the Gaussian normal distribution?

Consider some compact Riemannian manifold $M$. Fix some point $p$.
Consider a "sub-sphere of radius $r$" - i.e. set of points on distance $r$ from $p$.
Consider growth function $g(r)$ to be ...

0
votes

0
answers

87
views

### Using a theorem (which is originally set on 2-dim bounded domain in Euclidean space) on a torus

Actually I'm reading a paper on mean-field equation on torus by M.Struwe and G.Tarantello Here, they studied $$\tag{1}
-\Delta u=\lambda\left(\frac{e^u}{\int_{\Omega} e^u d x}-\frac{1}{|\Omega|}\right)...

0
votes

0
answers

49
views

### When considering an equation on flat torus, can we treat it as an equation on a square with opposite sites topologically identified? [migrated]

I need to use some theorems (which is originally for $2$-dim bounded domain in Euclidean space) on torus, what should I do to the equation? Can I just simply treat it as an equation on a square with ...

5
votes

3
answers

822
views

### Naturality of Lie bracket — alternate proof

Let $M$ and $N$ be smooth manifolds, and let $F: M \to N$ be a smooth map. Let $X$ and $Y$ be vector fields on $M$, and let $\tilde{X}$ and $\tilde{Y}$ be vector fields on $N$. We say that $X$ and $\...