All Questions
Tagged with dg.differential-geometry riemannian-geometry
1,985 questions
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Total curvature of a conjugate minimal surface
Let $s: S \to \Bbb R^3$ be an immersed minimal surface with finite total curvature and a proper annular end (possibly with other types of ends).
What is exactly meant by a proper annular end? It is an ...
- 11
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1
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208
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Frobenius theorem and the size of integral manifold
Let $X =(X_0,X_1)\in \mathbb{R}^2$ and $Y=(Y_0,Y_1)\in \mathbb{R}^2$ be two vector fields on $\mathbb{R}^2$ such that $X,Y$ are independent on each tangent plane and
$[X,Y]:=XY-YX=0$.
Then by ...
- 328
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1
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464
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Bochner Laplacian in coordinates
Sorry if this is a too basic question, but I didn't find an answer anywhere:
The connection Laplacian, or Bochner Laplacian, is the differential operator acting on $k$-tensor fields $T\in\Gamma^{\...
- 1,171
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58
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Smoothness of the Fréchet Function
Let $M$ be a compact Riemannian manifold and $d$ be the induced distance function. Suppose $\mu$ is a probability measure on $M$ with continuous density. The Fréchet function is defined as
$$
F(x) = \...
- 73
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126
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On the linearized evolution equations in general relativity
The following puzzles me already for quite some time: In mathematical relativity, especially in the discussion of the Cauchy problem, one usually works in the so-called ADM-Formalism, in which one ...
- 1,429
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208
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Riemannian structure on connected Hilbert manifolds
The infinite-dimensional separable Hilbert space $H$ has the unusual property that it is diffeomorphic to its unit sphere $S^{\infty}$. Therefore, $H$ admits the round metric as a complete and bounded ...
- 1,543
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157
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Can every surface be realized as a mean convex hypersurface in $\mathbb{R}^3$?
I'm wondering if every closed surface can be realized as a mean convex hypersurface in $\mathbb{R}^3$, i.e. the mean curvature vanishes or points inward.
Categorizing by genus: for $S^2$ ($g = 0$) ...
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Conformal Killing fields satisfy a third order PDE
Let $(M,g)$ be a $n$-dimensional Riemannian manifold with smooth metric $g$.
Proposition 3.2 in the paper "The Boost Problem in General Relativity" by O'Murchadha and Chistodoulou claims ...
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3
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How to get to the earliest time zone?
You are in a plane at some point on Earth. You want to be at the earliest time zone possible at the end of your flight. What is the optimal path to take?
Formally, fix spherical coordinates $(\theta, \...
- 6,195
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411
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Parallel transport on a vector bundle : expansion of the correspondance between normal and tubular coordinates
Let $(M, g^{TM})$ be a Riemannian manifold of dimension $n$. Let $X \subset M$ be a submanifold of dimension $n$ with boundary $\partial X$. Then we have a splitting of the tangent bundle
$$TM \vert_{\...
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Does anyone recognize this condition on a Riemannian metric on a vector space?
In the course of studying some oscillatory integral problems, the following strange condition came up. Let $V$ be a finite-dimensional real vector space. Let us say that a smooth Riemannian metric $...
- 114k
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123
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Asymptotics on the number of diffeomorphism classes in the Cheeger finiteness theorem
A result of Cheeger says that, given any even dimension and any $\delta > 0$, there are only a finite number of diffeomorphism types of compact simply-connected manifolds of that dimension which ...
- 3,212
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2
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626
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Elliptic regularity on manifolds: Is this true?
Let $(M,g)$ be a Riemannian manifold (without boundary) and denote by $\Delta_{g}$ the Laplace-Beltrami operator (or any other elliptic operator if you wish). I was trying to find a reference for the ...
- 1,171
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1
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128
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Is the heat kernel of a manifold $p$-integrable?
If $M$ is a separable, oriented Riemannian manifold, without any other assumption on its geometry, and $h$ is its heat kernel, it is known that $h(t,x,\cdot)$ is both integrable, and square-integrable ...
- 5,407
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2
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228
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A triangle comparison in CAT(0) spaces
Let $pxy$ be a triangle in a CAT(0) space $X$, and $p' x' y'$ be a triangle in $\mathbf{R}^2$ such that the lengths $|px|=|p'x'|$, $|py|=|p'y'|$ and the angle $\angle(xpy)=\angle(x'p'y')$. Let $z\in ...
- 7,149
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166
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Parallel transport of global sections and Riemannian curvature
A, perhaps, naive question from an algebraist/combinatorialist teaching differential geometry. Originally asked on math.SE but didn't receive a single comment in 3 days.
Consider a (real) smooth ...
- 3,513
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91
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Measurability of the union of cut loci along a curve
Let $(M,g)$ be a Riemannian symmetric space and $\alpha(s)$ be a geodesic. Define
$$
U(t)=\cup_{s\in[0,t]}{\rm Cut}(\alpha(s))
$$
as the union of the cut loci ${\rm Cut}(\alpha(s))$ along the curve $\...
- 125
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2
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589
views
How to learn intrinsic torsion
I want to learn about G-structure and intrinsic torsion. But I can find no textbook that details it.
If you can give me a reference about it, it would be much appreciated.
- 111
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183
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Osculating sphere at point of maximal curvature lies to one side
I'm looking for the higher dimensional version of this post, which says that given a curve $\gamma \subseteq \mathbb{R}^2$, the osculating circle will lie to one side of the curve at points of maximal ...
- 337
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1
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160
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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
$$
g(t)=\...
- 125
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343
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Clarifying a result of Klingenberg
I am looking for some clarification on a result of Klingenberg. For context, I have a complete Riemannian manifold for which I would like to compute a lower bound on the injectivity radius at each ...
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173
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$L^{p}$ estimate for $\Delta|\nabla u|$ on a manifold with bounded Ricci curvature
This is a more advanced question than Estimates of $\Delta|\nabla u|$ for harmonic function $u$ .
The Bochner formula and refined Kato inequality tells us that on a Riemannian manifold $(M^{n},g)$, if ...
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357
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Estimates of $\Delta|\nabla u|$ for harmonic function $u$
The well-known Bochner formula and refined Kato inequality (for harmonic function) tells us that for a harmonic function $u$ in $B_{2}\subset\mathbb{R}^{n}$,
$$
\frac{1}{n}|\nabla^{2}u|^{2}\leqslant |\...
- 155
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0
answers
104
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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.
...
- 169
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202
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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 ...
- 1,219
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82
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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 ...
- 4,081
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137
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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 ...
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153
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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 ($x^a$ ...
- 250
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165
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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) ...
- 1,097
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128
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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)...
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2
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266
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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 ...
user515818
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102
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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 ...
- 419
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85
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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 ...
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291
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Lie algebra cohomology of the space of vector fields
For a (closed and oriented) manifold $M$, the first Lie algebra cohomology $H^1(\mathrm{Vect}(M),C^\infty(M))$ of the space of vector fields with coefficients in smooth functions is isomorphic to $H^1(...
- 985
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158
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Conformal Killing vector fields on manifolds that are not asymptotically flat
Let $M = [1,\infty) \times S^2$.
Equip $M$ with the metric $g = dr^2 + r^2 (\gamma + h)$ where $\gamma$ is a metric on $S^2$ and $h$ is a $(0,2)$ tensor on $M$ that satisfies
$$h = O(1/r),\quad \...
- 969
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122
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The rigidity of $2$-dim sphere with constant sectional curvature in $\mathbb{R}^n$ for $n> 3$
If there is a smooth isometric embedding $f: (S^2, g)\rightarrow \mathbb{R}^n$, where $(S^2, g)$ is a sphere with Riemannian metric such that the corresponding sectional curvature is equal to $1$, and ...
- 799
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1
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85
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Jacobi fields in singular metric on quotient space
Consider the square $\Omega = (0,\pi) \times (0,\pi/2) \ni (r,\theta)$ endowed with the Riemannian metric
\begin{equation}
f^2 \big(\mathrm{d} r^2 + \sin^2(r) \, \mathrm{d} \theta^2 \big),
\end{...
- 5,038
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Does any warped product metric with harmonic Weyl curvature admit a structure of zero radial Weyl curvature?
A Riemannian manifold $(M, g)$ has harmonic Weyl curvature iff its Schouten tensor is Codazzi, and if there exists $f: M \to \mathbb{R}$ such that $W(\bullet, \bullet, \bullet, \nabla f) = 0$, one ...
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260
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Exterior differential systems on compact three-manifolds and Cartan-Kähler theory
Let $M$ be a compact three-manifold. I am interested in the following equation on $M$:
$ d e^i = \sum_{j,k}^3 \Theta^i_{jk} \, e^j\wedge e^k\, , \qquad i =1,2,3$
together with the following condition:...
- 2,816
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112
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Does any warped product metric admit a function with hessian proportional to the metric?
It is known that the existence of a function with hessian proportional to the metric implies that the metric is a warped product metric. Is the reciprocal true as well? I.e, if $(B \times N, g = g_B + ...
- 659
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110
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Moser iteration epsilon-regularity for non-linear system in general dimension
I am attempting to prove the following result in general dimension $n$. Given $(M^n,g)$ a Riemannian manifold with $\mathrm{Ric}_g \geq -(n-1)$ and $\mathrm{Vol}_g(B_1(x)) \geq v > 0$ for all $x \...
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0
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135
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Conformal laplacian on asymptotically flat manifolds with boundary
Let $g$ be an asymptotically flat metric on $M = \mathbb{R}^3 \setminus B_1$ where $B_1$ is the unit ball.
Suppose $X$ is a smooth vector field on $M$ that is decaying exponentially and satisfies
$$\...
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162
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The existence of a positive Green function for the Laplacian on $\mathbb R$
One can show explicitly and easily that the function $G(x,y) = \frac 1 2 |x-y|$ is a positive Green function for the Laplacian $\frac {\mathrm d ^2} {\mathrm d x ^2}$ on $\mathbb R$ (endowed with the ...
- 5,407
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200
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Bounded covariant derivative of curvature tensor
Let $M$ be a complete Riemannian manifold.
Suppose that there are positive constants $i_0$ and $K$ such that the injectivity radius of $M$ is at least $i_0$ and $|\mathrm{Rm}|\le K$ and $|\nabla \...
- 45k
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1
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255
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How do we calculate the gradient of this function defined using the Riemannian logarithm on a Riemannian manifold?
We consider the following function $\psi$ on an open subset $V\subset M,$ a Riemannian manifold of dimension $m,$ so that $\exp_p:U\to V$ is a diffeomorphism with its inverse $\log_p: V\to U$. Let $v\...
- 1,467
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1
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186
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Convex hull of 3 points in Cartan-Hadamard manifolds
Can the convex hull of $3$ points in a Cartan-Hadamard manifold be smooth?
A Cartan-Hadamard manifold $M$ is a complete simply connected manifold with nonpositive curvature (so it includes the ...
- 7,149
2
votes
2
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148
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Is any globally-hyperbolic manifold conformally equivalent to one with complete slices?
Let $(M,g)$ be a globally-hyperbolic Lorentzian manifold. By the seminal work of Geroch and Bernal-Sánchez, we know that
$$M=\mathbb{R}\times\Sigma,\,\,\,\quad g=-\beta^{2}dt^{2}+h_{t}$$
where $\Sigma$...
- 1,429
1
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1
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204
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Curve length in the Sasaki metric
I am trying to read Appendix II.A.2 (Distances in the tangent bundle) in Canary, Epstein, Marden (eds.), Fundamentals of Hyperbolic Manifolds: Selected Expositions and am stumbling over a calculation ...
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0
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113
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What is known about warped product metrics satisfying conditions more general than conformal flatness?
In this paper, the authors characterize warped product metrics which are conformally flat (the fibers must have constant sectional curvature, on some cases there is a limitation on the number of ...
- 659
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1
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162
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Going from piecewise to genuine geodesic without decreasing number of intersections?
Let $(M^2,g)$ be a complete, two-dimensional Riemannian manifold be given; also given is $\gamma: [0,\infty) \to M$, an injective geodesic in $M$.
Suppose there are two geodesic segments $\gamma_i : [...
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