Questions tagged [dg.differential-geometry]
Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.
8,645
questions
2
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0
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166
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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, ...
1
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0
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63
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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 ...
1
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0
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55
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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 ...
3
votes
1
answer
139
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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-...
1
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1
answer
148
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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)=\...
1
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0
answers
95
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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 ...
7
votes
0
answers
181
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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&...
5
votes
1
answer
223
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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 ...
3
votes
0
answers
161
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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 ...
4
votes
0
answers
90
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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$)...
7
votes
1
answer
304
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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 |\...
3
votes
3
answers
451
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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, ...
1
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0
answers
78
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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 $\...
0
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0
answers
96
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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?
1
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1
answer
72
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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 ...
0
votes
0
answers
69
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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}...
2
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0
answers
78
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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 ...
1
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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\...
4
votes
1
answer
212
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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 ...
16
votes
1
answer
713
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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 ...
1
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0
answers
125
views
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 ...
0
votes
0
answers
89
views
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 ...
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 ...
1
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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 ...
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 $\gamma_t:[0,1]\to M$ ...
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.
...
0
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0
answers
132
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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(\...
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 ...
0
votes
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 ...
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 ...
2
votes
1
answer
132
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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{...
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,...
0
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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 ...
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 $H$-...
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 ...
1
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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 ($x^a$ ...
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 ...
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) ...
2
votes
0
answers
116
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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
$...
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 ...
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)...
1
vote
2
answers
223
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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 ...
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 ...
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 ...
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\...
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 ...
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 ...
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 ...
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 ...
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 ...