All Questions
188 questions
1
vote
1
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258
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Isoperimetric inequality for domains in the exterior of a precompact open set in Riemannian manifold
Fix $n\geq 2$ and let $$\mathbb{H}^{n}=\mathbb{R}_{+}\times \mathbb{S}^{n-1}$$ be the hyperbolic space, so that any point $x\in \mathbb{H}^{n}$ can be represented in polar coordinates $x=(r, \theta)$, ...
- 14.4k
1
vote
0
answers
138
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References Request: Bach tensor
Recently I want to study about Bach tensor in detail. From the fundemental definition and properties to conformal invariant. Is there any references to me about Bach tensor? If there are some origins ...
- 11
2
votes
2
answers
523
views
Orthogonal smooth vector field on a Riemannian manifold
Consider a compact Riemannian manifold $M$ with a smooth metric, and a smooth vector field $X$ on $M$. My question is, when can we construct another smooth vector field $Y$ on $M$ such that $Y$ is ...
- 44.7k
6
votes
2
answers
317
views
Quasi-isometric embedding of graphs in non-compact riemannian surfaces
Given a complete riemannian surface $(S,m)$, where $S$ is homeomorphic to $\mathbb{R}^2$, I would like to find a weighted graph $G$ (which means a graph with real non-negative weights on the edges), ...
- 1,926
0
votes
0
answers
51
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References for local distance approximation over Riemannian manifolds [duplicate]
Over a complete Riemannian manifold $(M,g)$, in a neighborhood of $p \in M$, the local distance can be approximated as follows: $\forall v,u \text{ unit vectors in } T_pM, \text{ and small } s, t$
$$ ...
- 31
3
votes
1
answer
369
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Closed manifolds of nonnegative curvature operator are symmetric spaces
In an online webinar, I heard (not directly) the statement that (closed) manifolds of nonnegative curvature operator $\mathcal{R}\geq 0$ are symmetric spaces. Is this a valid theorem? Any reference ...
- 4,195
0
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1
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108
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Intersection Grassmanian planes
I am reading a paper that used Grassmanian planes properties. In particular, they studied the intersection of Grassmanian planes; they check the intersection Grassmanian of $n-k$-planes and ...
- 26.3k
6
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1
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229
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Does $\pi_k(M)\neq 0$ implies $\operatorname{ind}(\gamma) < k$?
Cross post from MSE. and sorry if this is an obvious question.
Here is a line of proof of Theorem 1.15 from
Brendle, Simon, Ricci flow and the sphere theorem, Graduate Studies in Mathematics 111. ...
- 4,195
1
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1
answer
243
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Reference for non-parallel harmonic $k$-forms
I want to get some deep understanding on closed orientable Riemannian manifolds admitting $k$-forms ($k\geq 2$) $\omega$ that satisfices the following conditions:
$$\nabla \omega\neq 0,\quad \Delta\...
- 108k
7
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0
answers
248
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Does the Hodge decomposition hold for equivariant differential forms?
Let $M$ be a Riemannian manifold. The Hodge decomposition tells that
$$
\Omega^*(M) = \mathrm{im} \ d \oplus \mathrm{im} \ d^* \oplus \mathscr H^*(M)
$$
where $d^*$ is the adjoint operator of the ...
- 2,789
8
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3
answers
1k
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Higher derivatives than Jacobi fields
The first and second derivatives of the distance function (either the full $d:M\times M\to \mathbb{R}$ function or the $d(p,\cdot):M\to \mathbb{R}$ function) as well as the derivative of the ...
- 12.9k
6
votes
1
answer
160
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Metrics of non-negative sectional curvature on $S^7$-bundles over $S^8$
In Curvature and symmetry of Milnor spheres, Grove and Ziller construct metrics of non-negative sectional curvature on $S^3$-bundles over $S^4$ (by using a cohomogeneity one action). In the same paper,...
- 6,546
4
votes
1
answer
503
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singular metric (with essential singularity)
Working on some $Q$-curvature equation in dimension $4$, I have been faced with singular metric of the form $(\mathbb{B}, e^{-1/\vert x\vert ^2} \vert dx\vert)$. I try to figure out to what those ...
- 914
10
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1
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3k
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Taylor expansion of the metric tensor in the normal coordinates
I am looking for a reference with a Taylor expansion of the metric tensor in the normal coordinates.
The coefficients should be written in terms of $\mathrm{Rm}, \nabla\mathrm{Rm}, \nabla^2\mathrm{Rm},...
- 108k
1
vote
0
answers
213
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Injectivity radius bounds for Riemannian manifolds of low regularity
In their seminal paper, Jeff Cheeger, Mikhail Gromov, and Michael Taylor derivated bounds on the injectivity radius of Riemannian manifolds with bounded sectional curvature of the form:
$
inj(p)\geq r ...
2
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0
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141
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For a 1-parameter family of metrics, how do we compute the derivative of the intrinsic geometrical objects like curvature, Hessian, etc
Consider a family of metrics and functions $(g(t), u(t))$ on $M:= \mathbb{R}^3 \setminus B_1$ satisfying
$$ g(0) = g_0, \quad g'(0) = \tilde g, \quad u(0) = u_0, \quad u'(0) = \tilde u$$
where $g_0$, $...
- 969
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5
answers
2k
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Reference request: Recovering a Riemannian metric from the distance function
Let $M = (M, g)$ be a Riemannian manifold, and let $p \in M$.
Writing $d$ for the geodesic distance in $M$, there is a function
$$
d(-, p)^2 : M \to \mathbb{R}.
$$
This function is smooth near $p$. ...
- 722
8
votes
1
answer
856
views
Are there mistakes in Kovalev's "Twisted connected sums and special Riemannian holonomy"?
This is kind of a strange and vague question... sorry about that.
I am really interested in $G_2$ Twisted Connected sums as described in this paper: https://arxiv.org/abs/math/0012189 "Twisted ...
- 188k
3
votes
0
answers
247
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Fibre metrics on non-linear bundles
Usually what is meant under a fibre metric is that one is given a (smooth) vector bundle $\pi:Y\rightarrow X$, and on each fibre $Y_x$ an algebraic inner product $g_x$ that varies smoothly from point ...
- 2,792
0
votes
0
answers
126
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mean curvature for codimension $>1$?
The mean curvature of a hypersurface in a Riemannian manifold is defined to be the trace of the second fundamental form. I was curious, does the notion of mean curvature generalise to higher ...
- 3,625
6
votes
2
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706
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Reference request: uniformization theorem
I would appreciate if someone could point me to some introductory literature/resources where I can learn about Poincaré's uniformization theorem at a basic level.
Any good powerpoint notes, short ...
- 63.9k
2
votes
0
answers
67
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Closed-form expression for Riemannian exponential maps on symmetric spaces
Besides the Poincaré model for the hyperbolic disc, $S^n$, and Euclidean space, what are known instances of a symmetric space $M$ (finite-dimensional) for which the exponential map is known in closed-...
- 63.9k
1
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1
answer
94
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Initial value problems on manifolds around submanifolds (reference)
I am looking for a reference on initial value problems formulated on smooth manifolds with initial conditions on submanifolds. More precisely, let $X$ be a smooth manifold and $Y\subset X$ a embedded ...
- 108k
15
votes
2
answers
2k
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Riemannian manifold as a metric space
I am looking for a reference to the following simple statement; it must be classical. (It is easy to proof, but I want to have a reference.)
A metric space $X$ that corresponds to a Riemannian ...
- 45k
3
votes
0
answers
188
views
References and results for the eigenvalues of Ricci tensor
I am looking for references or results that gives estimates for every eigenvalue of the Ricci tensor. For example, the least eigenvalue is related to the minimum of the Ricci curvature, what can we ...
- 1,774
4
votes
1
answer
301
views
Injectivity radius of parallel hypersurfaces
Let $(M,g)$ be a Riemannian manifold and let $N$ be a compact hypersurface isometrically embedded into $M$ and let $\eta$ denote a choice of unit normal vector field on $N$. It is then true that $N$ ...
- 3,223
4
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0
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880
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Scalar curvature in terms of second fundamental form, reference request
I would like to cite a reference for the following formula for scalar curvature:
If $\Sigma$ is a hypersurface in Euclidean space, then $R=H^2-\lvert A\rvert^2$, where $R$ is the scalar curvature ...
10
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1
answer
707
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Injectivity radius of manifolds with boundary
This question stems from the discussion in:
how to define the injectivity radius of manifolds with boundary?
Suppose $(M,g)$ is a compact Riemannian manifold with boundary. In this context, let ...
- 409
8
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1
answer
673
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Classification of compact globally symmetric spaces
It is known that any connected compact Lie group $G$ is a finite quotient of the product of a compact simply connected semisimple Lie group $\tilde{G}$ and a torus $\mathbb{T}^n$ (see for example ...
- 858
1
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1
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331
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Can divergence free vector fields be approximated by smooth ones?
If $M$ is a compact Riemannian manifold, is the space of $C^{\infty}$ divergence-free vector fields dense in the space of $C^r$ divergence-free vector fields, in the $C^r$ topology ($r\geq 1$)? How ...
- 52.3k
7
votes
2
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396
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Is every metric uniformly close to a metric with negative scalar curvature?
Let $M$ be a smooth manifold with non-empty boundary.
Let $g$ be a smooth Riemannian metric on $M$. Is the following true?
For every $\epsilon >0$ there exist a Riemannian metric $g_{\epsilon}$ ...
- 4,123
2
votes
0
answers
134
views
Hypersurfaces whose unit normal $N$ satisfies $[N,X] =0$ for every tangent vector field $X$
Let $M$ be a hypersurface of a Riemannian manifold, and assume that $M$ satisfies the following property:
For each $p \in M$, given a unit normal vector field $N$ defined in a neighborhood $U$ of $...
- 985
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0
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159
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The Laplacian of a tubular neighborhood
Let $(M,g_M)$ be a compact submanifold of $\mathbb{R}^n$. Are there any known results relating the spectrum of the Laplace-Beltrami operator of M to the spectrum of the Laplace-Beltrami operator of a ...
- 409
6
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0
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122
views
Given the Ricci decays rapidly to 0 at infinity, is the metric asymptotically flat?
Consider the manifold $M=\mathbb{R}^3 \setminus B$ where B is the ball with radius 1. Let $f \in C^{ \infty}(M) $ satisfying:
$$f = \frac{C(\theta, \phi)}{r} + O( r^{-2}) $$
Where $(r,\theta,\phi)$ ...
- 969
3
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0
answers
336
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Understanding Calabi's conjecture proof: What is it meant by the logarithm of a differential form?
I'm reading several books and articles concerning Yau's proof of the Calabi conjecture. I want to have a deep understading of how and why such proof actually works, but most articles are aimed at ...
- 153
11
votes
2
answers
2k
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Retraction of a Riemannian manifold with boundary to its cut locus
This question is edited following the comment of Joseph. He pointed out that the main object of the first version of this question is the cut locus.
Recall that the cut locus of a set $S$ in a ...
- 28.9k
5
votes
1
answer
594
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Existence of nonvanishing Killing field
Let $(M,g)$ be a closed Riemannian manifold.
Q Is there any research about the existence of nonvanishing Killing field, especially the nontrivial example.
- 6,995
1
vote
2
answers
1k
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Reference on Complex Geometry
For the preparation of a complex geometry lecture I am looking for a good literature. I already have standard literature like Huybrechts "Complex Geometry. An Introduction" and I am also using it. But ...
- 1,422
3
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1
answer
628
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Local Sobolev embedding on complete Riemannian manifold
Let $(M,g)$ be a complete Riemannian $m$-manifold, with bounded geometry and $m\geq2$. Suppose $Ric\geq(n-1)\kappa$. Let $B_p(r)$ be a geodesic open ball.
Q Can we find a constant $C=C(\kappa,r,m)$(...
- 28k
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0
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307
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Gradient estimate for Poisson equation on manifold
In Gilbarg-Trudinger's book 'Elliptic Partial Differential Equations of Second order', the maximum principle is used to derive the following gradient estimates for Poisson equations on Euclidean ...
- 307
10
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1
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403
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Positive Ricci curvature on fiber bundles
My advisor and I are working on Ricci curvature and an anonymous referee pointed out the following conjecture:
Let $F\hookrightarrow M\stackrel{\pi}{\to}B$ be a fiber bundle from a compact manifold ...
- 6,337
14
votes
3
answers
963
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Conjugate points on cut locus
Let $M$ be a Riemannian with nonempty boundary $\partial M$.
Define multiplicity of $x\in M$ as the number of minimizing geodesics from $x$ to $\partial M$.
The following fact seems to be standard:
...
- 45k
2
votes
0
answers
66
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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 ...
- 1,915
11
votes
1
answer
529
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Length decreasing homotopies of curves
Let $M$ be smooth compact riemannian manifold with boundary and $\varphi_0: S^1\to M$ be a rectifiable curve (or a smooth one). I would like to find a reference to the following statement:
Statement. ...
CommunityBot
- 1
6
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0
answers
355
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Higher order variations of Riemannian geodesics
Consider a mapping $\Gamma$ from the Euclidean plane or an open subset to a Riemannian manifold $M$ so that each $\Gamma(s,\cdot)$ is a geodesic.
There is a well established theory of the first order ...
- 8,086
64
votes
12
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22k
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Advanced Differential Geometry Textbook
I tried this post on StackExchange with no luck. Hopefully the experts at MathOverflow can help.
In algebraic topology there are two canonical "advanced" textbooks that go quite far beyond the usual ...
- 389
4
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0
answers
116
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$\ell_p$ geodesic distance on smooth Riemannian manifold and Logarithmic Sobolev Inequalities
Bear with me, I'm not a professional geometer. Recently, I've been studying Logarithmic Sobolev Inequalities (LSI) for probability distributions on manifolds (e.g as done in works of Bobkvo et al. ...
- 6,853
2
votes
1
answer
239
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Projection of a ball in the ambient space to a manifold
Let $B_h (x)$ be the ball of radius $0<h \ll 1$ centered at $x\in \mathbb{R}^d$.
Let $I=[0,1]^{d-1}$ be the unit cube in $\mathbb{R}^{d-1}$, and let $f:I \to \mathbb{R}$ be a $C^2$ function. Then $...
- 17.2k
6
votes
1
answer
399
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A possible generalization of the exponential map
Let $M$ be a $n$-dimensional Riemannian Manifold, fix $p\in M$, and $1<k<n$. Do we know if the following is true?
For any $k$-dimensional subspace $V$ of $T_p M$, there exists a minimal ...
- 2,299
12
votes
1
answer
3k
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how to define the injectivity radius of manifolds with boundary?
For manifolds without boundary one defines the injectivity radius as the maximal radius where the exponential map is a diffeomorphism. One can then show that the injectivity radius is the maximum ...
- 5,407