Questions tagged [dg.differential-geometry]

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

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Nonlinear Poisson brackets associated with nilpotent (matrix) Lie algebras?

With every finite-dimensional Lie algebra $\mathfrak{g}$ one can associate a linear Poisson bracket on $\mathfrak{g}^\ast$. With some more restrictions on $\mathfrak{g}$ and some extra ingredients, ...
Ricardo Buring's user avatar
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Looking for examples of 3rd-order contact transformations

In the Herglotz Lectures on Contact Transformations and Hamiltonian Systems, after going through contact transformations of the form $X=X(x,y,p)$, $Y=Y(x,y,p)$, $P=P(x,y,p)$ it is stated: As a final ...
Eli Bartlett's user avatar
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Complex geodesic coordinate, local ramified map, and the conic metric

Remark: I have asked this question in MSE, however, I got no responses. This is the reason I come to ask here. I am looking forward to some advices. Thanks in advance Let $X$ be a compact Kaehler ...
Invariance's user avatar
3 votes
1 answer
139 views

Request for non-Einstein positive constant scalar curvature Kähler surfaces

I am curious about concrete examples of compact cscK manifolds in complex dimension two, in particular cscK surfaces with positive scalar curvature. There are of course the Fano (del Pezzo) Kähler-...
Garrett Brown's user avatar
1 vote
1 answer
148 views

Differentiability of an integral of geodesic distance

Let $(M,g)$ be an $m$-dimensional Riemannian symmetric space and $d(\cdot,\cdot)$ be the geodesic distance function. Fix any $\alpha\in M$ and $v\in T_\alpha M$ with $\|v\|=1$. Q1: Define $$ g(t)=\...
Hengchao Chen's user avatar
1 vote
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How causal is a strongly causal purely electric spacetime?

Take a generic Lorentzian spacetime $(M, g)$ where $M$ is a time-oriented 4d manifold and $g$ is the Lorentzian metric that is strongly causal and purely electric. According to this answer: Is every ...
Bastam Tajik's user avatar
7 votes
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181 views

Are the spaces BG for compact connected groups G ind-projective or ind-Kähler?

Let $G$ be a compact connected group, or maybe better its complexification. By thinking about the simplicial Borel space, or using $n$-acyclic $G$-spaces for higher and higher $n$, it's "easy&...
Ben Webster's user avatar
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1 answer
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Clarifying a result of Klingenberg

I am looking for some clarification on a result of Klingenberg. For context, I have a complete Riemannian manifold for which I would like to compute a lower bound on the injectivity radius at each ...
E G's user avatar
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$L^{p}$ estimate for $\Delta|\nabla u|$ on a manifold with bounded Ricci curvature

This is a more advanced question than Estimates of $\Delta|\nabla u|$ for harmonic function $u$ . The Bochner formula and refined Kato inequality tells us that on a Riemannian manifold $(M^{n},g)$, if ...
Xin Qian's user avatar
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Exponential map for tangent space of space of distributions $\mathscr{P}_2(X)$

In Chapter 8 of the book Gradient Flows In Metric Spaces and in the Space of Probability Measures by Ambrosio et al., the tangent space to the space of distributions on $X$ (let's say $X=\mathbb{R}^d$)...
Juno Kim's user avatar
7 votes
1 answer
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Estimates of $\Delta|\nabla u|$ for harmonic function $u$

The well-known Bochner formula and refined Kato inequality (for harmonic function) tells us that for a harmonic function $u$ in $B_{2}\subset\mathbb{R}^{n}$, $$ \frac{1}{n}|\nabla^{2}u|^{2}\leqslant |\...
Xin Qian's user avatar
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3 answers
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Difference in length of two dimensional concentric closed paths

Two bicyclists ride side by side around a smooth loop with the outside bicyclist a distance of $D$ from the inside bicyclist. How much further does the outside bicyclist ride? If the loop is a circle, ...
Terry Wagner's user avatar
1 vote
0 answers
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Can the volume of a neighborhood of the cut locus be arbitrarily small?

Let $(M^n,g)$ be a complete, $n$-dimensional Riemanian manifold without boundary, maybe non-compact. Let $p\in M$ be a point, and $C_p$ the cut locus. It's known that $C_p$ has Hausdorff dimension $\...
mathmetricgeometry's user avatar
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Torsion free Chern connections and Kähler manifolds

Let $(M,h)$ be an Hermitian manifold and let $\nabla$ be the associated Chern connection. Is it true that $(M,h)$ is Kähler if and only if $\nabla$ is torsion free?
Zoltan Fleishman's user avatar
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1 answer
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When is the real Abel-Jacobi-Albanese map injective?

$ \def\tMA{\tilde{M}_a} \def\tomega{\tilde{\omega}} \def\tx{{\tilde{x}}} \def\tzeta{\tilde{\zeta}} \def\T{{\mathbb T}} \def\R{{\mathbb R}} \def\Z{{\mathbb Z}} \def\raw{\rightarrow} $ I want to work ...
Philip Boyland's user avatar
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Existence of covering space with trivial pullback map on $H^1$

I have seen somewhere the following claim (but can't remember where): let $M$ be a connected orientable closed smooth manifold with $b_1(M)=1$, then there exists a connected covering space $p:\tilde{M}...
F. Müller's user avatar
2 votes
0 answers
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On the convergence of isometries

Proposition: Let $(M,g)$ be a closed, compact Riemannian manifold. Let $f_n:M\to M$ be a sequence of isometries satisfying $$ \|f_n - \iota\|_{L^2} \le \frac{1}{n}, $$ where $\iota:M\to M$ is the ...
Raz Kupferman's user avatar
1 vote
0 answers
133 views

$L^{p}$ estimate for $\frac{|\nabla f|^{2}}{f}$

I’m trying to obtain an $L^{p}$ estimate under certain conditions. Suppose $f\in C^{2}(B_{2})$ is a function satisfying $0<f<1$ and $f\Delta f>\frac{1}{n} |\nabla f|^{2}$, where $B_{2}\subset\...
Xin Qian's user avatar
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Let $G \subset \mathrm{GL}(n)$ be a matrix Lie group. Does there exist an affine connection under which the matrix and manifold exponential coincide?

Let $G\subset \operatorname{GL}(n)$ be a matrix Lie group. I am curious about curves $\gamma(t) = g \exp(tv)$, where $g \in G$, $v \in \mathfrak{g}$, and $\exp(.)$ is the matrix exponential. If we ...
Spencer Kraisler's user avatar
16 votes
1 answer
713 views

The determinant as a differential operator

According to Gårding, the determinant is a hyperbolic polynomial over the space $\mathbf{Sym}_n$ of real symmetric $n\times n$ matrices. More precisely, it is hyperbolic in the direction of the ...
Denis Serre's user avatar
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Poincaré-Hopf Theorem for domains with a point of vanishing curvature

Consider $\Omega \subset \mathbb{R}^2$ a convex planar domain having positive curvature on the boundary except for a point $p \in \partial \Omega$ where the curvature vanishes. I would like to know ...
DrHAL's user avatar
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Bounding the area of the image of a set by product of maximum of lengths

Let $F:[0,1]\times[0,1]\to \mathbb {R}^2$ be a smooth function. Given $x\in [0,1]$, let $\ell_x:=\{x\}\times [0,1]$, and given $y\in [0,1]$, let $\ell_y:=[0,1]\times \{y\}$. My question feels ...
JustSomeGuy's user avatar
5 votes
1 answer
387 views

A question about the existence of spin maps

Let $M, N$ be two smooth manifolds, not necessarily spin. My question is the following: How can we construct a non-constant spin map $f:M\to N$ of degree zero? Here spin map means that $f$ preserves ...
Radeha Longa's user avatar
1 vote
0 answers
98 views

Site structure on smooth fibered manifolds

Let $\mathsf{FB}_{s}$ be the category whose objects are smooth fibered manifolds, and whose morphisms are smooth strong projectable maps. Recall that given fibered manifolds $(\pi:Y\rightarrow X)$ and ...
Bence Racskó's user avatar
3 votes
0 answers
63 views

Continuous trajectory of midpoints of length-minimizing geodesics

Let $M$ be a smooth Riemannian manifold, $x$ be a point in $M$, and $\lambda:[0,1]\to M$ be a continuous path. Can we find a family of length-minimizing constant speed geodesics $\gamma_t:[0,1]\to M$ ...
Cave Johnson's user avatar
2 votes
0 answers
87 views

Convergence of diffeomorphisms

Let $(\Sigma, g)$ be a compact $n$-dimensional Riemannian manifold without boundary. Let $F_i$ be a sequence of diffeomorphisms of $\Sigma$ and $u_i$ be a sequence of positive scalar functions. ...
user486255's user avatar
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132 views

Relative bounds for vorticity

Write the vorticity equation as \begin{equation}\label{Eq20} \begin{split} \dfrac{\partial}{\partial t} v_i & = \biggl[|\textbf{v}|~|\nabla u_i|\cos(\beta_i)- |\textbf{u}|~|\nabla v_i|\cos(\...
free_lancer's user avatar
0 votes
1 answer
114 views

How to estimate the distance between geodesics and points for Riemannian manifold with positive sectional curvature

Assume that $ M $ is a complete Riemannian manifold and there exists $ k>0 $ such that $ K(q)\geq k $ for any $ q\in M $, where $ K $ is the sectional curvature of $ M $. Let $ \gamma $ be a closed ...
Luis Yanka Annalisc's user avatar
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0 answers
269 views

Space which is diffeomorphic to CP^2 # -CP^2

The manifold $\mathbb{CP}^2 \# -\mathbb{CP}^2$, the non-trivial $\mathbb{S}^2$ bundle over $\mathbb{S}^2$, is known to be diffeomorphic to the space that we will now describe. Represent $\mathbb{S}^3 ...
Dimpi Paul's user avatar
3 votes
0 answers
59 views

Conformal group equals isometry group for locally homogeneous manifolds

$\DeclareMathOperator\Conf{Conf}\DeclareMathOperator\Iso{Iso}$Let $ M $ be a locally homogeneous Riemannian manifold, in other words the universal cover $ \tilde{M} $ has a transitive action by the ...
Ian Gershon Teixeira's user avatar
2 votes
1 answer
132 views

Boundedness of an exit time from a campact set

Let $n\geq 1$ and $v\in\mathcal{C}^1(\mathbb{R}^n,\mathbb{R}^n)$. For $x_0\in\mathcal{O}$, let $\big(x(t)\big)_{t\geq 0}$ be the solution of \begin{align*} & x(0)=x_0 \\ & \dot{x}=v(x). \end{...
G. Panel's user avatar
  • 557
2 votes
0 answers
51 views

Lefschetz operator on bundle-valued forms

For a holomorphic vector bundle $V \rightarrow X$ endowed with a Hermitian structure, one may define the corresponding Dolbeault-like operators $\bar{\partial}_V: \Omega^{p,q}(V) \rightarrow \Omega^{p,...
Eweler's user avatar
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0 votes
0 answers
105 views

Determining Metrics from Scalar Curvature

I am delving into the intricate relationship between metrics and scalar curvature in Riemannian geometry. My objective is to understand the feasibility and methods for solving the inverse problem of ...
lming2's user avatar
  • 45
8 votes
0 answers
296 views

Flat Maurer-Cartan connection iff flat Berry connection

I am studying two connections on induced representation spaces $\text{Ind}_{H}^{G} \Gamma$, where $H \subseteq G$ are groups, and $\Gamma$ is an irrep of $H$. The first is the canonical or $H$-...
Victor V Albert's user avatar
2 votes
0 answers
59 views

Is the Leray projection continuous with respect to the Frechet topology of smooth periodic vector fields in $3$ dimensions?

Let $\mathbb{T}^3:=(\mathbb{R}/\mathbb{Z})^3$ be the $3$-torus and $C^\infty(\mathbb{T}^3,\mathbb{R}^3)$ be the Frechet space of smooth periodic vector fields on $\mathbb{T}^3$. By Helmholtz ...
Isaac's user avatar
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1 vote
0 answers
131 views

Torsion free connection $\implies$ Jet coordinates $=$ Taylor expansion coefficients?

Suppose we have some smooth n-dimensional manifold $M$ endowed with basis 1-forms $\theta^a$ with $a=1\cdots n$. Then $\theta^a$ are sections of the coframe bundle $F^* M$. In local coordinates ($x^a$ ...
R. Rankin's user avatar
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5 votes
1 answer
218 views

Commutative/ symmetric second covariant derivative

Consider a smooth manifold $M$ together with an affine connection (or covariant derivative) $\nabla$ on the tangent bundle $TM$. Is it possible to have an affine connection, possibly with non-zero ...
Khaled T.'s user avatar
2 votes
0 answers
151 views

A property of the Riemannian metric on the symmetric space $X_\textsf{d}=\textsf{SL}(d,\Bbb R)/\textsf{SO}(d)$

Consider the symmetric space $X_\textsf{d}=\textsf{SL}(d,\Bbb R)/\textsf{SO}(d)$. For any $M\in X_d$, by transporting the Killing form on $T_I(X_d)$ (the space of symmetric matrices with trace zero) ...
Random's user avatar
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2 votes
0 answers
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The double quotient of SU(N) by its diagonal maximal torus

$\DeclareMathOperator\SU{SU}$The special unitary group $\SU(N)$ contains $T^{N-1}$ as a maximal torus, which we take to be the diagonal subgroup of $\SU(N)$. Can we describe the double quotient space $...
Yilmaz Caddesi's user avatar
0 votes
1 answer
150 views

Formulating a 3D "twist" transformation for a unit circle into a lemniscate while preserving arc length

I'm exploring the transformation of a 2D unit circle into a lemniscate (infinity symbol) by fixing two antipodal points and "twisting" the circle (in the 3rd dimension) such that the ...
swami's user avatar
  • 369
0 votes
0 answers
110 views

How to build a representation of the diffeomorphism group of $U(n)$?

Given that $U(n)$ is a smooth manifold I would like to know if there is a way of building a representation of $\text{Diff}(U(n))$ once you pick a particular (finite dimensional) representation of $U(n)...
Nicolas Medina Sanchez's user avatar
1 vote
2 answers
223 views

Isometric embeddings of $\Bbb H^3$

Consider the upper-half space model of hyperbolic $3$-space $\Bbb H^{+}_{3}$, the unique, simply-connected, $3$-dimensional complete Riemannian manifold with a constant negative sectional curvature ...
user avatar
5 votes
0 answers
77 views

General questions on stochastic calc on manifolds

I've done coursework in differential geometry and stochastic calculus, but I haven't formally seen any connections between the two. I have read that both information geometry and stochastic ...
djr's user avatar
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1 vote
1 answer
109 views

Precise mathematical relation between chirality (or $\gamma_5$) and (spatial) orientation in $1+3$ Minkowski spacetime

This is a bit of a qualitative question, but I have great difficulty finding a reference that clarifies the point I have been confused about. So, I guess I need to ask here.. Let us restrict atttetion ...
Isaac's user avatar
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3 votes
1 answer
167 views

Precise definition of a linear total differential operator

In the works of A. M. Vinogradov on calculus on the infinite jet space, differential equations and "diffieties", a central notion is that of a $\mathcal C$-differential operator. If $\pi:Y\...
Bence Racskó's user avatar
2 votes
0 answers
103 views

Can closed forms be resolved to exact ones by a submersion?

Let's say that a smooth manifold $M$ is $k$-coverable if there exists a surjective smooth submersion $N \to M$ such that $H^k(N, \mathbb{R}) = 0$. For example, every manifold is $1$-coverable by the ...
Mattis Bakken's user avatar
0 votes
0 answers
121 views

Naming convention for different type of triangulations

When studying random geometries and related mathematical/physical stuff conflicting naming convention pops up regarding the naming of the different ensemble types of triangulations (in general ...
Kregnach's user avatar
0 votes
0 answers
82 views

Spectrum of Laplace-Beltrami operator on tensors

Let $(M, g)$ be a complete Riemannian manifold diffeomorphic to $\mathbb{R}^n$. Under appropriate geometric assumptions concerning the geometry near infinity, but without any curvature sign ...
Chris's user avatar
  • 389
1 vote
0 answers
75 views

Closed form ODE solutions for Jacobi field/eigenfunction of Laplacian on hyperbolic space

I'm trying to compute Jacobi fields of the hyperbolic disk $\mathbb{H}^m$ considered as a minimal hypersurface in $\mathbb{H}^{m+1}$ in the half model. References to literature or solutions to the ...
JMK's user avatar
  • 301
0 votes
1 answer
340 views

Relation between trivial tangent bundle $\Leftrightarrow$ certain characteristic classes of tangent bundle vanish [closed]

We know that framing structure means the trivialization of tangent bundle of manifold $M$. string structure means the trivialization of Stiefel-Whitney class $w_1$, $w_2$ and half of the first ...
zeta's user avatar
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