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Questions tagged [dg.differential-geometry]

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

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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 $...
Xin Qian's user avatar
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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 ...
Praphulla Koushik's user avatar
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 $(x, y) \in \mathbb{S}^2 \times \mathbb{S}^2$, ...
MathLearner's user avatar
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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?
Lefevres's user avatar
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 ...
Chevallier's user avatar
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 ...
Kiyoon Eum's user avatar
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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 ...
Ali's user avatar
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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 ...
B.Hueber's user avatar
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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 ...
user302934's user avatar
-1 votes
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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(...
Wrloord's user avatar
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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 $...
Johannes's user avatar
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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 ...
MyShepherd's user avatar
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 ...
pizzalberto's user avatar
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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 ...
Student's user avatar
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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 $x$ ...
Ivo Terek's user avatar
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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 ...
0207's user avatar
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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 (...
Olivier's user avatar
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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 ...
Y.Guo's user avatar
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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}...
Measure32's user avatar
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,...
Akerbeltz's user avatar
  • 506
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 ...
Radeha Longa's user avatar
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 ...
Learning math's user avatar
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 ...
Partha's user avatar
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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 ...
Bobby-John Wilson's user avatar
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 ...
Severin's user avatar
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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: ...
Bobby-John Wilson's user avatar
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?
Partha's user avatar
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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?
Jake Wetlock's user avatar
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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 ...
math_is_hard's user avatar
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?
Lefevres's user avatar
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 ...
user avatar
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 $n$-...
Yueqi's user avatar
  • 73
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 ...
Lefevres's user avatar
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 ...
James Dibble's user avatar
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 ...
George's user avatar
  • 143
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 ...
Partha's user avatar
  • 853
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, ...
ZZZ's user avatar
  • 11
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 ...
George's user avatar
  • 143
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 ...
HCH's user avatar
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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 ...
Student's user avatar
  • 655
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 ...
Paul Cusson's user avatar
  • 1,735
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 $\...
user815293's user avatar
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)...
stupid_question_bot's user avatar
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. ...
wsz_fantasy's user avatar
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 ...
Alex Bogatskiy's user avatar
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 ...
unknownymous's user avatar
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 ...
Alexander Chervov's user avatar
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)...
Elio Li's user avatar
  • 729
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 ...
Elio Li's user avatar
  • 729
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 $\...
Zhang Yuhan's user avatar

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