0
votes
0answers
65 views

Why Shape space is manifold?

In Shape analysis, often shape is considered as continuous parametrized closed curve and the shape space as Hilbert Riemannian manifold. Can any one help me to understand, why the shape space is ...
4
votes
3answers
139 views

Kernel of the Laplacian on a compact manifold

Is there a way to characterise the kernel of the Laplace-Beltrami operator on a compact manifold without boundary? Or is it just "the set of functions $u$ such that $-\Delta u = 0$?"
1
vote
0answers
37 views

Extension of a Pseudodifferential Operator

Let $M$ be a smooth manifold with countable atlas, and define the distributions $\mathscr{D}'(M)$ as the dual space to the smooth densities with compact support, and $\mathcal{E}'(M)$ as the dual ...
0
votes
0answers
68 views

Why are affine subspaces also sometimes called linear manifolds?

According to Wikipedia, an affine subspace is a subset of a vector space closed under affine linear combinations. That is, linear combinations whose scalar coefficients sum to 1. It's not clear to ...
2
votes
1answer
54 views

Is a non-compact Riemannian manifold a “measure space”?

One can define $L^p$ spaces for measure spaces with a given measure. Is a non-compact (i.e., it has a boundary) bounded Riemannian manifold a measure space? I am thinking of the manifold $(0,T) \times ...
1
vote
1answer
65 views

The completion of $C_c^\infty(M)$ with respect to $\lVert \nabla u\rVert_{L^2(M)}$ on a compact Riemannian manifold

Let $M$ be a compact smooth Riemannian manifold (eg. a smooth hypersurface) and let $X$ be the space given as the completion of $C_c^\infty(M)$ with respect to the norm $\left(\int_{M} |\nabla ...
7
votes
0answers
101 views

Showing L infinity norm bounded by L2 norm on a manifold

I have the following problem that I'm working on: Suppose $(M, g_{ij})$ is a compact Riemannian manifold. Assume $u$ is a smooth, nonnegative function which satisfies the differential inequality ...
1
vote
0answers
22 views

A question on the completely positive maps and manifold structure

I was reading a paper in which the curvature and Euler characteristic of a completely positive map (in finite dimensions). Let \begin{equation} \Phi(X)=\sum_{j=1}^nV_jXV_j^* \end{equation} be a ...
3
votes
0answers
45 views

Variational Problem on a manifold - bounding my functional from below

Let $(M, g_{ij})$ be a compact Riemannian manifold. Let $f \in C^{0,\alpha}(M)$ be a given function. I have the linear operator $L = \Delta h - 12u^2h$, where $u \in C^{2,\alpha}$. I need to show ...
0
votes
0answers
24 views

Derivative of function on parametrized manifold

Given $U\subset\mathbb{R}^d$. Let $\alpha:U\mapsto \mathbb{R}^k$ be an injective function such that $\alpha(U)$ is $d-$dimensional parametrized manifold. Now define $\beta:\alpha(U)\mapsto U$ by ...
0
votes
1answer
24 views

Derivative on parametrized manifold

Let $U\subset \mathbb{R}^m$ is open and $\alpha: U\mapsto\mathbb{R}^n$ is $m-$dimensional parametrized manifold with $m\leq n$. I have two following questions: If we are given $v\in \alpha(U)$, ...
1
vote
1answer
57 views

Few questions about global analysis relating $C^k$ functions

First question is about the definition. Let $U$ be an open subset of $R^n$. Let $f$ be $k$ times continuously differentiable function on $U$. $C^k$ norm of $f$ is defined as sum of uniform norm of ...
3
votes
1answer
176 views

Use of cut-off functions and partitions of unity

This is a simple problem, but I would still be very thankful if you could give me an advice on it. I'm trying to show that in a compact M-dimensional manifold, $$\int e^w \sqrt{g}\, dx \leq C \exp ...
1
vote
2answers
79 views

Distributional derivatives on hypersurface?

In a paper I was reading, the define a set $Q=(0,T)\times \Omega$, where $\Omega \subset \mathbb{R}^n$ is a bounded domain, and then they write $$\langle \frac{d}{dt}u - \Delta u, \varphi ...
0
votes
0answers
27 views

About a function space on $\bigcup_{t \in (0,T)}\Gamma(t) \times \{t\}$

For each $t$, let $\Gamma(t)$ be a $C^k$ hypersurface without boundary. Define $$Q = \bigcup_{t \in (0,T)}\Gamma(t) \times \{t\}.$$ I am trying to understand some properties of this space $Q$. ...
0
votes
1answer
19 views

About the boundary of a set of the form $Q_i = \bigcup_{t \in (0,T)}\Omega_i(t) \times \{t\}$

Let $\Omega$ be a bounded (open) domain. For every $t \in [0,T]$, let $\Omega_1(t), \Omega_2(t)$ be open subsets of $\Omega$, with $S(t)$ the interface separating $\Omega_1(t)$ and $\Omega_2(t)$. ...
1
vote
1answer
45 views

Lipschitz map between hypersurfaces/manifolds

if $A$ and $B$ are compact hypersurfaces or manifolds and $F:A \to B$ is a $C^1$ diffeomorphism, does it follow that $F$ is Lipschitz? I am think of the case where these hypersurfaces are boundaries ...
3
votes
0answers
74 views

Does a “typical” reproducing kernel on a manifold generate an infinite-dimensional RKHS?

Let $M$ be a finite dimensional manifold and $k:M\times M\rightarrow\mathbb{R}$ a smooth reproducing kernel. Is the set of all such $k$ with the property that the reproducing kernel Hilbert space ...
0
votes
2answers
95 views

Two definitions of integral on boundary $\int_{\partial\Omega}f$?

I have seen two definitions of an integral of a function $f:\partial\Omega \to \mathbb{R}$ from the boundary of an open set $\Omega \subset \mathbb{R}^n$ where the domain is Lipschitz. 1) ...
7
votes
2answers
81 views

Detecting compactness from the ring of smooth functions

Given a smooth manifold $M$, is there some ring-theoretic property (preferably not mentioning $M$) such that $C^{\infty}(M)$ has this property if and only if $M$ is compact?
3
votes
1answer
262 views

Question about theorem 3.2 from Morse theory by Milnor

The demonstration of the theorem 3.2 in the book Morse theory by Milnor THEOREM $\mathbf{3.2.}$ Let $f:M\to\bf R$ be a smooth function, and let $p$ be a non-degenerate critical point with index ...
2
votes
1answer
184 views

An other question about Theorem 3.1 from Morse theory by Milnor

In the proof of theorem 3.1 they put : $\langle X,\nabla f \rangle =X(f)$ after that they say that: for a curve $c$ on $M$ then $\left\langle\dfrac{\mathrm{d}c}{\mathrm{d}t},\nabla ...
3
votes
1answer
214 views

Submersion Theorem for Banach Spaces

I'm having difficulty proving a well-known result from functional analysis. Any hints would be greatly appreciated. Fix a Fréchet differentiable map of Banach spaces $g: X \to B$. Assume that, at a ...
0
votes
1answer
57 views

meaning of “doubly inward-pointing”

I am currently trying to understand the $b$-calculus developed by R. Melrose. an important part of the theory is the stretched product of a manifold $X$ with boundary $\partial X$. looking at the ...
2
votes
2answers
810 views

Does the Implicit mapping theorem imply the inverse mapping theorem?

Does the Implicit mapping theorem imply the inverse mapping theorem?
1
vote
1answer
145 views

Why is the laplacian positive-definite

Let (M,g) be compact Riemannian manifold (possibly $\partial M\neq\emptyset)$ Now I have read, that "the laplace-beltrami operator is a positive definite operator". I have shown, if M is a closed ...
0
votes
1answer
132 views

$L^2$ definition on a manifold

Is the space of $k$-forms on a compact Riemannian manifold $M$ with the inner product given by $$(\alpha,\beta)=\int \alpha \wedge *\beta=\int g(\alpha,\beta)dv$$ which is called «$L^2$ product in ...