All Questions
30 questions with no upvoted or accepted answers
9
votes
0
answers
336
views
Nash embedding for 3 manifolds
The Nash embedding theorem tells us that every smooth Riemannian m-manifold can be embedded in $R^n$ for, say, $n = m^2 + 5m + 3$ (edit: 14 is a better bound for compact 3 manifolds thanks @mme). What ...
- 4,081
5
votes
0
answers
219
views
Is the volume functional analytic in the space of embeddings? What about locally?
Let $(M^{n+1},g)$ be an analytic Riemannian manifold and let $\Sigma^n$ be a closed analytic manifold. Denote by $\operatorname{Emb}(\Sigma, M)$ the space of all smooth (or maybe analytic) two-sided ...
- 2,095
5
votes
0
answers
101
views
How is this product of tensors defined?
I am reading the paper “ The first eigenvalue of a small geodesic ball in a riemannian manifold”, by Karp and Pinsky, from where I took the following:
Here, $\Delta_{-2}$ denotes the usual Laplacian ...
- 2,095
4
votes
0
answers
334
views
Hodge decomposition on non-compact manifolds
Let $(\mathcal{M},g)$ be a compact Riemannian manifold without boundary. Then we have the well-known Hodge decomposition
$$\Omega^{k}(\mathcal{M})\cong\mathcal{H}^{k}(\mathcal{M})\oplus\mathrm{ran}(\...
- 1,171
4
votes
0
answers
196
views
Let $p : \tilde{M} \to M$ be the universal cover. Can we ever deduce curvature properties of $M$ from the curvature of $\tilde{M}$?
Let $M$ be a $C^{\infty}$-smooth, connected, paracompact manifold with universal cover $\tilde{M}$. Assume $M$ is not simply connected, so that the covering map $p : \tilde{M} \to M$ is not the ...
- 1,379
4
votes
0
answers
195
views
Classifying singularities of the Ricci flow
Context:
A solution $(M^n, g(t))$ of the Ricci flow is said to encounter a Type III Singularity if $g(t)$ is defined for all $t \geq 0$ and:
$$
\sup _{\mathcal{M}^{n} \times[0, \infty)} \|\...
- 659
4
votes
0
answers
116
views
$\ell_p$ 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
4
votes
0
answers
191
views
A quantity associated with a Riemannian surface
Assume that $E$ is a Riemannian vector bundle, then its structure group is reduced to $O(n)$. Then the structure group of $E \oplus E$ is reduced to $D(O(n) \oplus O(n)) \subset Sp(2n)$ ...
- 356
3
votes
0
answers
117
views
Geometric intuition behind definition of $\delta$-necklike points of the Ricci flow
In "The Ricci Flow: An Introduction", the authors define a $\delta$-necklike point of the Ricci flow as a point $(x, t)$ where $$\|\text{Rm} - R (\theta \otimes \theta)\| \leq \delta \|\text{...
- 659
3
votes
0
answers
109
views
Application of Santalo’s formula
Suppose that $(M,g)$ is a compact smooth Riemannian manifold with a smooth boundary and suppose that $f$ is a smooth function on $M$ with the property that
$$ \int_I f(\gamma(t))\,dt=0,$$
for any ...
- 4,143
3
votes
0
answers
101
views
Minimal normal graph
Let $(M^3,g)$ be a complete and orientable Riemannian $3$-manifold and let $\Sigma^2 \subset M$ be a compact orientable totally geodesic surface embedded in $M$. For $f \in C^{2,\alpha}(\Sigma)$ with ...
- 2,095
3
votes
0
answers
531
views
Geodesics (Local vs Global)
Let $M$ be a Riemannian manifold, and $p,q\in M$ two points. Now, if $M$ is a complete metric space the Hopf–Rinow theorem ensures that there is a geodesic $C\subset M$ joining $p$ and $q$ that ...
user75933
2
votes
0
answers
107
views
Finite dimensional manifolds as subspace of $\mathbb{R}^\mathbb{N}$
For embedded submanifold, specifically with ambient space being $\mathbb{R}^{n}$, there are many nice properties and results. Specifically there are many examples of matrix manifolds such as the ...
- 275
2
votes
0
answers
71
views
Domain of definition of a certain mapping
Suppose we have a compact smooth riemannian manifold $(M,g)$ and a $\mathcal{C}^2$ diffeomorphism $f$ of this manifold.
I am studying the mapping
$$\tilde{f}(x)= \exp_{f(x)}^{-1} \circ f \circ \exp_x \...
2
votes
0
answers
354
views
Continuity of surface integrals on level sets
Let $\phi:\mathbb{R}^2\to\mathbb{R}$ such that $\phi^{-1}(0)\neq\emptyset$ and $\phi\in C^1(W)$ where $W$ is a compact neighborhood of $\phi^{-1}(0)$, with $\nabla\phi\neq 0$ in $W$. So there is some $...
- 1,759
2
votes
0
answers
35
views
Can one extend a Hermitian bundle from a compact manifold with boundary to its Riemannian double?
Let $M$ be a compact Riemannian manifold with boundary, and let $E \to M$ be a Hermitian vector bundle, endowed with a compatible connection. Let $\tilde M$ be a Riemannian double of $M$.
Does $E$ ...
- 5,407
2
votes
0
answers
205
views
Can a non-compact manifold become compact by cutting it?
I'm trying to understand a step in a proof, where one starts with a non-compact manifold $V$ containing a trapped (2-sided, closed) surface $\Sigma$ that's non-separating. In order to complete the ...
- 121
2
votes
0
answers
137
views
Heat-Flow on continuous differential forms and the Feller peroperty
Let (M,g) be a complete Riemannian manifold. It is well known that the Laplace operator is essentially self-adjoint on $C^\infty_c(M)$. This extends to the (de Rahm) Laplace operator on forms. Thus in ...
2
votes
0
answers
216
views
A geometric rank of Riemannian manifolds
There are various ranks which have been assigned to a smooth manifold. The following ranks are two examples of such ranks:
The maximum number of global independent vector fields which can be defined ...
- 356
1
vote
0
answers
117
views
Question on globally hyperbolic manifolds and coordinates
Consider a globally hyperbolic Lorentzian manifold $(M,g)$. Then, a well-known result of Bernal-Sánchez (see Theorem 1.1 in arXiv:gr-qc/0401112) states that it can globally be written as
$$M=\mathbb{R}...
- 1,171
1
vote
0
answers
88
views
Metric of negative curvature on connected sum
Let $(M_1,g_1)$ and $(M_2,g_2)$ be two Riemannian manifolds of dimension $n\geq 2$. If we consider the connected sum $M=M_1\mathbin{\#}M_2$ of the two manifolds; can one get a smooth metric $g$ on $M$ ...
- 11
1
vote
0
answers
218
views
Is this generalization of differentiable manifolds to mixed dimensions a known object?
Suppose you are studying the evolution of some electromagnetic quantity in a conductor consisting of objects of several dimensions, i.e. wires, plates and balls.
This would amount to studying the ...
- 203
1
vote
0
answers
123
views
Subdividing a Compact Bounded Curvature Manifold into Charts with Bounded Lipschitz Constant
Let $M \subset \mathbb{R}^d$ be a compact smooth $k$-dimensional manifold embedded in $\mathbb{R}^d$. Let $\mathcal{N}(\epsilon)$ denote the size of the minimum $\epsilon$ cover $P$ of $M$; that is ...
- 71
1
vote
0
answers
54
views
Extension of a function from a subset of a manifold to the unit sphere
There is a line in this following paper (page no- 221, the paragraph before the Lemma 2.2)
Otsu, Yukio; Shiohama, Katsuhiro; Yamaguchi, Takao, A new version of differentiable sphere theorem, Invent. ...
- 930
1
vote
0
answers
225
views
Extending fibre metrics of submanifolds to Riemannian metrics
Let $M$ be a smooth manifold and $S\subseteq M$ a properly embedded smooth submanifold. Suppose that we have a fibre metric on $TM|_S$, i.e. a positive definite real inner-product on $T_pM$ for all $p\...
- 11
1
vote
0
answers
84
views
A problem of defining addition in a Quotient space
Let $\mathcal{C}$ be the space of all parametric curves $x:[0,1]\rightarrow \mathbb{R}^2$. Let the set of all re-parameterizations of curves is $\Gamma = \{\gamma : [0, 1] \rightarrow [0, 1]| \gamma (...
- 213
1
vote
0
answers
127
views
Riemann normal coordiantes and change of metric
Le $(M,g)$ be Riemannian manifold. Fix point $p\in M$. We can define the map
$$\exp: U \subset T_p M \rightarrow M$$
$$\exp(X) = \gamma_{p,X}(1)$$
where $t\mapsto \gamma_{p,X}(t)$ is geodesic such ...
- 552
0
votes
0
answers
466
views
Example metrics for exotic R4
I'm a physics student trying to understand what exotic manifolds, such as exotic R4, means. Is there known examples what the Riemannian metric of some exotic R4 (or some exotic sphere) would be? Does ...
- 113
0
votes
0
answers
90
views
The idealizer of the space of vector fields with vanishing divergence
The space of smooth vector fields on a manifold is counted as a Lie algebra with the usual Lie bracket structure.
Is there a Riemannian manifold of dimension at least $2$ which satisfies either of ...
- 356
0
votes
0
answers
236
views
Angle between two vectors in a Minkowski (Finsler) space
Given a Minkowski (or Finsler) space $(V,F)$, I am wondering how to define the angle between two vectors $w$ and $v$. I first thought it must be as $$\cos\theta(w,v)=\frac{g_w(w,v)}{\sqrt{g_w(w,w)g_w(...
- 227