# Questions tagged [dg.differential-geometry]

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

5,708 questions

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### Exponential decay of resolvent kernel

For the integral kernel of the Laplacian $\Delta$ on $\mathbb{R}^n$, consider the resolvent $R(\lambda) := (\lambda - \Delta)^{-1}$ and let $R(\lambda; x, y)$ be its kernel, which is a smooth function ...

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111 views

### Orthonormal vector fields on a Riemannian manifold

Let $(M,g)$ be a Riemannian manifold. We equip the tangent bundle $TM$ with the Sasaki metric $g_s$.
Assume that $X: M \to TM$ is a vector field on $M$.
We say that $X$ is an orthonormal vector field ...

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97 views

### Variation Formula of APS $\eta$-Invariant and Chern-Simons Theory

In Perturbative Expansion of Chern-Simons Theory with Noncompact Gauge Group, the author proved the variation formula of the APS $\eta$-invariant. I have a few questions about their proof. I will ...

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49 views

### Mapping to distorted constant Gauss curvature surfaces of revolution

There are three questions here. We imagine a flexible membrane that is scrolled out so as to straighten it.
1) How can we find a surface isometrically mapped from a surface of constant negative Gauss ...

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114 views

### Global symplectic reduction

Let $M$ be a symplectic manifold equipped with a hamiltonian action of a compact Lie group $G$ with moment map $\mu\colon M\to \mathfrak g^*$. Assume $c\in \mathfrak g^*$. Then the symplectic ...

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216 views

### Prescribing a gradient direction

Let $\Omega= \{(x,y) : \frac{1}{2} \leq x^2+y^2 \leq 1\}$ and $S = \{(x,y) : x^2+y^2 = 1\}$ the unit circle, and $X=w^{1.\infty}(\Omega;\mathbb{R})$ the space of Lipschitz valued functions. We denote ...

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113 views

### The Hausdorff dimension of the union of singular orbits and exceptional orbits

Suppose we have a compact connected Lie group $G$ acting as isometries on a compact manifold $M^n.$ Then is it necessarily true that the Hausdorff dimension of the union of singular and exceptional ...

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236 views

### Lie group actions on $S^n$ with some invariant hypersphere but no totally geodesic ones

Does there exist a compact connected Lie group $G$ acting smoothly as isometries on the standard sphere $S^n$ for some $n\ge 3$, so that no totally geodesic hypersphere $S^{n-1}$ is $G$-invariant, but ...

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241 views

### Gaussian curvature of a surface does not take the constant value 1?

I came across this very complex equation (calculating the Gaussian curvature of a surface):
\begin{align*}
1 \not\equiv &-\frac{m}{2}\Bigl(\frac{3}{2}C+Su^{-1}-Tu^{-1}+2+Qu^{-1}\Bigr)\\
&\...

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110 views

### Lifting a diffeomorphism into a spinor bundle automorphism

I know several papers that treat this, but it seems that most of these papers do things very differently with quite different conclusions, so I am confused.
Basically, when one tries to do classical ...

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105 views

### Can scalar curvature and diameter control volume? Round 2

This is a follow up to a question by Yiyue Zhang. Can scalar curvature and diameter control volume?
The original question asked whether scalar curvature bounds and small diameter bounds were enough ...

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95 views

### Does an 8 dimensional compact Riemannian manifold contain an embedded minimal hypersurface?

It is well know that if $(M^{n+1},g)$ is a compact Riemannian manifold and $n \leq 6$ then there exists a smooth, embedded minimal hypersurface $\Sigma^n$ in $M$ (infinitely many such $\Sigma$, even)
...

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330 views

### What is known about sufficient conditions for the rigidity of a convex surface?

A convex surface is a connected open subset of the boundary of a convex body in $\mathbb{R}^3$.
An "infinitesimal bending" of a convex surface $S$ is a deformation of $S$ given by a velocity field $v:...

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**1**answer

453 views

### Physical (GR) Differential Geometry?

I am looking for problem lists or books which contain open problems in the area of mathematics motivated by physics. Ideally, I am looking for questions asking about which reduce to some calculation ...

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133 views

### A geometric rank of Riemannian manifolds

There are various ranks which have been assigned to a smooth manifold. The following ranks are two examples of such ranks:
The maximum number of global independent vector fields which can be defined ...

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119 views

### Holomorphic map, Instantons of Complex Projective Space and Loop Group

It seems that holomorphic (or rational) maps play a crucial role to relate the following data:
Instanton in 1-dimensional complex Projective Space $$\mathbb{P}^1$$
in a 2 dimensional (2d) spacetime.
...

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120 views

### Convexity of curves in Riemannian surfaces

It is known that a curve $f:[0,2\pi]\to \mathbf{R}^2$ is convex if $\partial_t (\arg f'(t))\ge 0$. My question is: does this statement have an analogue in the setting of Riemannian surfaces instead of ...

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76 views

### geodesic balls in the conformal change

Consider a compact smooth Riemannian manifold $(\mathcal{M}, g)$. We consider a conformal change of the metric. Let $\tilde{g}=\phi g$, where $\phi$ is smooth and positive. Moreover, it satisfies $...

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101 views

### Polygon of convex arcs

Convex polygons in the plane $R^2$ arise in linear programming where the constraints are linear. The objective linear function attains its maximum at a vertex of the feasible region(if exists).
Assume ...

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113 views

### Minkowski isometries

Consider theorem 1.7 from chapter III of 'Elementary differential geometry' by O'Neill. It says that:
Theorem 1.7: If $\phi$ is an isometry of $E^3 $, then there exists a unique translation $T$ and a ...

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91 views

### Sectional curvatures under simple maps

Suppose that we have a submanifold $X$ of $\mathbb{R}^n$ with the induced Euclidean metric, whose sectional curvatures we have a handle of (say, they are lower bounded by some $\kappa$).
Is there a ...

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180 views

### Foliation with a compact leaf

Let $M$ be a closed oriented manifold, and $F$ be a fixed foliation of $M$. We assume the dimension and codimension of $F$ are both greater than $1$.
Q Under what condition, we can say that $F$ ...

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45 views

### Decomposition of a Jacobi field along a lightlike geodesic

Consider a Lorentzian manifold of dimension $1+n$ (with $n\geq1$) and a lightlike geodesic $\gamma(t)$ on it.
One can define a Jacobi field $J(t)$ along $\gamma$ in the usual way without issues.
In ...

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299 views

### When $\int_M \exp(-d_M(x,y)^2/t) dvol(y)$ becomes constant for a Riemannian manifold $M$?

Let $(M,g)$ be a closed and connected Riemannian manifold. $d_M$ is its geodesic metric and $dvol_M$ is its standard volume measure. For each $t>0$, define a map $f:M\rightarrow\mathbb{R}_{>0}$ ...

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217 views

### Log-concavity of areas of level sets

Suppose $f: \mathbb{R}^d \to \mathbb{R}$ is a smooth convex function.
Consider the level sets of the function, namely $M_s = \{x: f(x) = s\}$.
Is it true/known that the surface areas of $M_s$ are ...

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157 views

### effectively distinguishing knots

It was proven, I think by Mijatović EDIT: by Waldhausen, that there is an effective algorithm for distinguishing knot complements (the effective constants were found by Coward and Lackenby). The bound,...

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63 views

### Some hypersurface has a positive second fundamental form potentially

Notation : $r^2=x^2+y^2$.
Exercise : Define $$F_\sigma (x,y)= (f_\sigma
(x,y),\frac{x^2-y^2}{2},xy)$$ where $$f_\sigma (x,y)=(1- \sigma^2
r^2)(x-\sigma x^3,y-\sigma y^3)$$
Define $ G_\sigma: \mathbb{...

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127 views

### A variation on four-vertex theorem

Is it true that, if a closed, strictly convex curve has exactly four vertices (extrema of curvature), then any circle has at most four points of intersection with it?

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91 views

### Why might the Lawson minimal surface $\xi_{1,2}$ have a Morse index 9?

In page 15 of the article New Applications of Min-max Theory, Andre Neves said that: a wishful thinking would suggest the Lawson minimal surface $\xi_{1,2}$ in the 3-sphere to have a Morse index 9, ...

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88 views

### Weak elliptic maximum principle on manifolds without strict ellipticity

This question is not to be confused with the similarly titled question here.
In the above lined question, I gave a complete answer, but noticed that things are apparently not so simple in the ...

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172 views

### Do eigenfunctions determine the geometry of a manifold? If so, do finitely many suffice?

Let $X$ be a smooth, Riemannian manifold. It is known that the geometry of $X$ can be recovered from its heat kernel $k_{t}(x,y)$, using Varadhan's Lemma: $\displaystyle\lim_{t \to 0} t \log k_{t}(x,y)...

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210 views

### Lorentzian analogue to Thurston geometries

Is there an analogue to the eight Thurston geometries for Lorentz metrics?
If so, how many "disctinct" geometries are there in the Lorentzian case?
And which closed 3-manifolds admit metrics which ...

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152 views

### Towards recognizing St. Venant geometrical invariant

Using partial derivative notation we can express Gauss curvature $K$ in cartesian coordinates:
$$\quad p= \partial w/ \partial x, q= \partial w/ \partial y; r=\frac{\partial ^2w}{{\partial {x} ^2} },...

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229 views

### Weinstein neighborhood theorem for Lagrangians with Legendrian boundary

$\require{AMScd}$
Weinstein's neighborhood theorem says that every Lagrangian has a standard neighborhood. The more precise statement goes like this.
Theorem 1: (Lagrangian Neighborhood Theorem) ...

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29 views

### Existence of smooth principal direction near umbilics

Given a hypersurface $M^n\hookrightarrow \mathbb{R}^{n+1}$, $n\geq 3$, and a point $p\in M^n$ does there exist a smooth orthonormal frame of principal directions in a neighbourhood of $p$?

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316 views

### Is the set of Lorentzian metrics metrizable?

Fix a differentiable non-compact manifold $M$. Denote by $\mathrm{Lor}(M) := \{\text{Lorentzian metrics on $M$}\}.$ One can define a topology on this set via: fix any open covering $\mathcal{A}$ on $M$...

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74 views

### Classification of Euclidian-like Klein geometries in spirit of Erlangen program

All we know about Erlangen Program approach to geometry. The main idea is to consider Lie Group $G$ acting transitively on smooth manifold $M$ (e.g. nLab). I really like such style of thinking, but ...

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80 views

### How much area can I see on average if I’m hovering over a deformed sphere?

I had asked this question on math.stackexchange (https://math.stackexchange.com/questions/2958307) but did not receive an answer.
I’m interested in getting some estimates for how much area I can see ...

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90 views

### Ricci flow with surgery without the “no locally separating $\Bbb RP^2$” assumption

In many places, Ricci flow with surgery is done with orientable manifolds. Morgan and Tian do not require orientability, but instead they impose the condition that $M^3$ have no embedded $\Bbb RP^2$ ...

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151 views

### What kind of locally symmetric space is a rational sphere

Using Dehn Surgery, we can construct compact hyperbolic $3$-manifolds with vanishing Betti numbers $b_1=b_2=0$, i.e., a rational homology $3$-sphere.
My question is the following.
Is there other ...

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242 views

### Principal curvatures of $\mathbb{R}^{n^2}$-embedded SO(n)

It's well known that the sectional curvatures of a Lie group, endowed with a left-invariant metric have a nice closed-form formula $k(X,Y) = \frac{1}{4} \|[X Y]\|^2$.
I'm wondering if the following (...

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221 views

### Weak parabolic maximum principle on Riemannian manifolds

I'm studying by myself Mean Curvature Flow and I'm reading the paper "Interior estimates for hypersurfaces moving by mean curvature" by Klaus Ecker and Gerhard Huisken, specifically, I'm reading the ...

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88 views

### What is a category of “Lepagean equivalent” or “variation problem”?

I get to know about it form Mark Gotay's work An exterior differential system approach to the Cartan form, in that paper he defined the canonical Lepagean equivalent. The following is cited from it:
...

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253 views

### Is this a 2-cyclic cocycle ? Does it have a nontrivial geometric interpretation?

Let $S$ be a surface in $\mathbb{R}^3$.
Inspired by the $2$-cyclic cocycle defined in page 20 of the book "Non-commutative geometry" by Alain Connes, we consider the following $3$-linear map on the ...

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170 views

### Almost complex structure and intrinsic torsion

Given a $2m$-dimensional manifold $M$, an almost complex structure $J$ is equivalent to a $\text{GL}(m,\mathbb C)$-structure on $M$.
I wonder why the intrinsic torsion of the $\text{GL}(m,\mathbb C)$...

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45 views

### Morse index of a closed minimal surface with a small disc removed

Consider the following observation:
Let $c_1$ be a geodesic on the unit round sphere $S^2$ with length $2\pi-\epsilon$, where $\epsilon$ is sufficiently small. Then $c_1$ has Morse index one ...

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161 views

### Diffeomorphism type of Ricci-flat four manifolds

Let $(M,g)$ be an irreducible compact and simply connected Ricci-flat Riemannian four-manifold. My first questions are as follows:
A) Is there a classification of the possible homeomorphism types of ...

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50 views

### One-parameter group of nonvanishing vector field

Let $M$ be a smooth manifold( if necessary one can assume it is closed), $V$ be a non-vanishing vector field of $TM$.
Q: Under what condition, we can say that $\overline{\exp(tV)}$, i.e. the closure ...

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47 views

### Foliation of cylinders by constant mean curvature spheres

Let the cylinder $S^{n-1}\times \mathbb R$ be equipped with the standard metric $g$. Suppose that there exists a sequence of metrics $g_{\epsilon}$ on $S^{n-1}\times \mathbb R$ such that $g_{\epsilon}$...

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117 views

### Dimensions of the instanton moduli space from Atiyah-Hitchin-Singer

Atiyah-Hitchin-Singer Ref 1 states that the number of
virtual dimensions of the instanton moduli space
for SU(N) Yang-Mills theory with topological charge $\mathcal{Q}$ over a manifold $X$ is given ...