All Questions
12 questions
2
votes
1
answer
224
views
The differentiability of the distance function on asymptotically flat manifolds
Let $M = \mathbb{R}^3 \setminus \overline{B_1}$ where $\overline{B_1}$ is the closed unit ball.
Let $g$ be an asymptotically flat metric of the form $g_{ij} = \delta_{ij}+h_{ij}$ in standard ...
- 969
1
vote
1
answer
149
views
Completeness of asymptotically Euclidean manifolds
Say that you place an asymptotically Euclidean metric on $\mathbb{R}^3,$ e.g. $\mathbb{R}^3$ is endowed Riemannian metric $g$ such that $\text{supp}(g^{ij}-\delta^{ij})\subseteq\{|x|\leq R\}$ for some ...
- 13
1
vote
1
answer
331
views
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 ...
- 19
3
votes
0
answers
159
views
Upper bound on the geodesic distance in a Lipschitz domain
I was wondering if the following result is true. If yes, could you please suggest a reference. The result seems to have been used at several papers without quoting any reference. Is the proof ...
- 895
2
votes
1
answer
239
views
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 $...
- 3,574
9
votes
1
answer
630
views
$C^{k,\alpha}$ diffeomorphisms and vector fields
This feels like something I should know, but I have a hard time finding a definite reference.
Let $M$ be a compact (Riemannian) manifold, $k\ge 1$ be an integer and $\alpha\in(0,1)$. When v is a $C^k$...
- 14.4k
5
votes
1
answer
267
views
Generalized Plateau problem with non-Jordan boundary
Let $C_\pm$ be the two circles obtained by intersecting the cylinder $x^2+y^2=R^2$ with the planes $z=\pm 1$, on which we mark four points $A_\pm:(R,0,\pm 1)$ and $B_\pm:(-R,0,\pm 1)$. Assume that $R$...
- 2,581
12
votes
1
answer
2k
views
Sard's Theorem For Banach Spaces
Given a smooth map from $\phi: B \rightarrow M$ where $B$ is a Banach Space and $M$ is a finite dimensional smooth manifold (for example, the end point map for a control system), what is the strongest ...
- 2,099
2
votes
1
answer
255
views
Parameter dependent differential equation in a Lie group
It is well-known that a linear differential equation in a finite-dimensional vector space depends continuously on some external parameters (for details see below). I search for an explicit reference ...
- 5,824
3
votes
3
answers
2k
views
How do we use an Ehresmann connection to define a semispray?
Let $M$ be a differentiable manifold, let $TM$ be its tangent bundle, and consider $TTM$, the double tangent bundle.
Let $V \subseteq TTM$ denote the vertical subbundle, which is determined in a ...
- 8,512
4
votes
0
answers
162
views
Question about density of $C^{\infty}(M,N)$ in $W^{1,p}(M,N)$ with $N$ not compact
Hi!
Let $M$ be a compact manifold possibly with boundary with $\dim(M)=m$, let $N$ be a non compact manifold with $\dim(N)=n$. Let me recall the definition of the sobolev space $W^{1,p}(M,N)$. ...
- 1,727
7
votes
4
answers
3k
views
How does curvature change under perturbations of a Riemannian metric?
Let $M$ be a compact subset of $\mathbb R^2$ with smooth boundary, and let $g$ be a Riemannian metric on $M$. If $g'$ is another Riemannian metric which is "close" to $g$, then they should have ...
- 8,512