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Non-unital algebras in geometric algebra, smooth envelopes

In Nestruev's (2000) Smooth Manifolds and Observables, the authors define an $\mathbb{R}$-algebra as a commutative, associative algebra with unit (p. 21). A natural generalization of this definition ...
supergeneric's user avatar
6 votes
1 answer
390 views

Equivariant implicit function theorem

Let $f:\mathbb{R}\times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ be a smooth function and $G\subset \operatorname{SO}(n)$ be a $1$-dimensional compact Lie group (diffeomorphic to the circle). ...
Ben's user avatar
  • 63
4 votes
0 answers
115 views

Delta distributions that are smooth on strata of a singular manifold

This is a mild reformulation of a previous question. Let $R = C^\infty(\mathbb{R}^N)$ and let $I$ be an ideal in $R$ which cuts out an $n$-dimensional "singular $C^\infty$ manifold $X$" in $\...
Dmitry Vaintrob's user avatar
3 votes
2 answers
247 views

Morse approximation with bounded number of critical points

Let $(M^3,g)$ be a compact Riemannian 3-manifold and let $f\in C^{\infty}(M)$ be a smooth function. Does there exist a constant $k>0$ (possibly depending on $M$ and $g$) such that $f$ can be $C^2$-...
cork_twist's user avatar
11 votes
1 answer
560 views

Different smooth structures on the infinite jet bundle (for the purposes of calculus of variations)

Let $\pi:Y\rightarrow X$ be a (smooth, finite dimensional) fibred manifold. Since no other fibrations will be considered on $Y$, I will identify $(Y,\pi,X)$ with $Y$. The finite order jet bundles are ...
Bence Racskó's user avatar
1 vote
0 answers
78 views

Asymptotics of eigenvalues of first-order self-adjoint elliptic operators

Let $D$ be a first-order self-adjoint elliptic operator on a closed Riemannian manifold $M$. Then $D$ has discrete spectrum in $\mathbb{R}$, and there is an orthonormal basis for $L^2(M)$ consisting ...
geometricK's user avatar
  • 1,903
3 votes
1 answer
109 views

Identification of smooth operators with rapidly decreasing matrices

In a paper I was reading, it was mentioned that if $M$ is a closed Riemannian manifold, then by fixing a basis for $L^2(M)$ consisting of eigenfunctions of the Laplacian, the space of smoothing ...
geometricK's user avatar
  • 1,903
3 votes
0 answers
164 views

Extension of normal vector field to a domain

Let $\Omega \subset \mathbb R^3$ be a bounded regular simply connected domain contained in a ball $S$. Assume also that $\Omega$ is simply connected by surfaces (i.e. every regular closed surface ...
user934318's user avatar
2 votes
1 answer
120 views

Smooth dependence in the fixed point theorem between complete Fréchet manifolds

Let $X,Y$ be complete metric spaces, and let $\Sigma:X\times Y\rightarrow Y$ be a continous mapping which satisfies the following property: there exists a $C<1$, such that for all $x\in X$ and $y_{...
MyShepherd's user avatar
5 votes
1 answer
406 views

Connection on a Hilbert bundle

Is there a well-defined notion of connection on a measurable bundle of Hilbert spaces?
Hasib's user avatar
  • 103
6 votes
2 answers
436 views

What properties should $C(M,\mathbb{R})$ have when $M$ is a $n$-dimensional manifold?

Let $M$ be a n-dimensional manifold, $C(M,\mathbb{R})$ be the function space of continuous function from $M$ to $\mathbb{R}$. What kind of properties should $C(M,\mathbb{R})$ has, to reflect the ...
Ken.Wong's user avatar
  • 523
4 votes
2 answers
335 views

If $\Omega$ is locally Lipschitz, then $\Omega = \bigcup_{k = 1}^N \Omega_k$ for $\Omega_k$ star shaped with respect to an open ball $B_k$

I am reading Galdi's Introduction to the mathematical theory of Navier Stokes equations and there is an argument which comes up quite often that I really don't understand. In many theorems of Chapter $...
Falcon's user avatar
  • 452
1 vote
0 answers
38 views

Extending the Dirac operator on an open subset of a manifold and preserving positivity

Let $M$ be a spin manifold and $U\subseteq M$ an open ball. Let $D$ be the Dirac operator on $M$ with respect to some Riemannian metric $g$, acting on sections of the spinor bundle $S\to M$. Suppose ...
geometricK's user avatar
  • 1,903
12 votes
1 answer
1k views

Smoothness of distance function to a compact set

Fix a non-empty compact subset $K\subseteq \mathbb{R}^n$ and let $d_K(x):=\min_{z \in K} \,\|z-x\|$ be the map sending any $x\in \mathbb{R}^n$ to its distance from $K$. Suppose that: $K$ is regular : ...
ABIM's user avatar
  • 5,405
0 votes
1 answer
147 views

The relationship between the first eigenfuntions and the second eigenfuntions on sphere [closed]

Recently I considered the following question: If we give a second eigenfuntions $g$ on sphere, then can we construct a first eigenfuntions $f$ by $g$? Is there any relationship between the first ...
管山林's user avatar
7 votes
1 answer
266 views

Visualizing the wave operator in two dimensions

For $n\geq 1$, let $D_n$ be the Dirac operator on the spinor bundle on the $n$-dimensional sphere $S^n$. For example, $D_1$ acts on the trivial bundle $S^1\times\mathbb{C}\to S^1$, and can be ...
geometricK's user avatar
  • 1,903
4 votes
0 answers
143 views

Sobolev space of maps between manifolds with boundary

Let $(M,g)$ and $(N,h)$ be compact Riemannian manifolds with non-empty smooth boundary. If we consider the Sobolev space $W^{1,p}(M,N)$, is there a reference on how to model this as a manifold? If ...
Somnath Basu's user avatar
  • 3,423
2 votes
1 answer
1k views

Weak derivatives and Sobolev spaces on Riemannian manifolds

I am decently experienced on Sobolev spaces on Euclidean spaces, but I just know basic ideas on Riemannian manifolds and want to understand something on Sobolev spaces on them. Let $(M,g)$ be smooth ...
Marko Rajkovic's user avatar
16 votes
2 answers
1k views

Examples of Banach manifolds with function spaces as tangent spaces

I have recently been learning the theory of Banach manifolds through Serge Lang's book on Differential Manifolds. So far the objects seem rather interesting but my intuition always comes from the ...
proba_124's user avatar
  • 161
3 votes
1 answer
1k views

Friedrichs mollifiers and Sobolev spaces

$\renewcommand{\epsilon}{\varepsilon}$The following is from John Roe's book Elliptic operators, topology and asymptotic methods. $S$ is a vector bundle on a compact manifold $M$, but I think for my ...
Carlos Esparza's user avatar
3 votes
1 answer
190 views

Laplace eigenfunction on a polygonal domain symmetric about an axis

Consider a polygon $\Omega \subseteq \mathbb{R}^2$, and let us consider the usual Laplacian operator $\Delta = \partial_x^2 + \partial_y^2$, with Dirichlet boundary conditions. My question comes from ...
user170399's user avatar
3 votes
0 answers
198 views

Dirichlet to Neumann operator and the Riesz transform

Consider the manifold $M := \mathbb{R}^3 \setminus B_1$ where $B_1$ is the unit ball. Equip $M$ with an asymptotically flat metric $g$ of high order. Let $\gamma$ be the induced metric on $\partial M$....
Laithy's user avatar
  • 969
-1 votes
2 answers
129 views

Is it possible for all of the smooth/continuous curves in $R^3$ to form a Hilbert space? [closed]

Under which condition can it form a Hilbert space? Or what space can it form? You can write down certain condition to make it to be a Hilbert space, e.g., Let $$p(t)=[x(t),y(t),z(t)]^T\in \text{R}^3$$ ...
Nan Zhang's user avatar
3 votes
0 answers
73 views

The diversity of Riemannian metrics adapted to a given foliation، A Krein Millman view point(2)

Inspired by this answer to the linked question we add a more bounded conditions to this post. This question is asked seperately because the previous one had a complete answer so we did not revise ...
Ali Taghavi's user avatar
5 votes
1 answer
204 views

The diversity of Riemannian metrics adapted to a given (1 dimensional) foliation, A Krein Millman view point

Let $X$ be a Kronecker vector field on the two dimensional torus $\mathbb{T}^2$. Let $K$ be the space of all 1- forms $\alpha$ of class $C^1$ on $\mathbb{T}^2$ which satisfy $d\alpha=0,\;\alpha(X)=1$...
Ali Taghavi's user avatar
2 votes
0 answers
382 views

Poincaré inequality holds on Riemannian manifolds (min max principle)

In YuChang Xia's book "Eigenvalues on Riemannian Manifolds" Page 4 equation (1.16)Poincaré inequality: I want to know which manifold(s)/function(s) can make the inequality hold. What if we ...
Grantsome's user avatar
6 votes
1 answer
338 views

Atiyah-Singer for Riemannian and Kaehler manifolds

I am trying to understand the proof of the Atiyah--Singer index theorem, and would like to see how it works for two "simple" examples. Could somebody direct me to a proof for the special ...
Dick Johnson's user avatar
3 votes
0 answers
125 views

Green operator of elliptic differential operator and radius of convergence

Let $E \to X$ be a hermitian vector bundle over a compact Kähler manifold and let $L$ be a self-adjoint elliptic linear differential operator on $E$. Suppose that $E \to X$ and $L$ are real-analytic. ...
Simon Parker's user avatar
  • 1,383
6 votes
0 answers
532 views

Hamiltonian dynamics on cotangent bundle

I'm stuck with the following claim made in Section 13.1 of Y-G. Oh's book "Symplectic topology and Floer homology". Assume that $N$ is a differential manifold and $S_0 ,S_1\subseteq N$ two ...
TheWildCat's user avatar
6 votes
1 answer
509 views

Path integral as quantum mechanics on the tangent bundle

Let $X$ be a configuration space, a finite-dimensional manifold. By "quantum mechanics on $X$" I mean a linear evolution equation on complex-valued functions on $X$, determined by a ...
Dmitry Vaintrob's user avatar
1 vote
0 answers
235 views

Fredholmness of elliptic operator on Hölder spaces

Let $(M,g)$ be a smooth oriented closed Riemannian manifold, $E\to M$ a smooth vector bundle, and $C^{k,\alpha}(E)$ the Banach space of sections of $E$ that are $k$-times differentiable (with respect ...
user158773's user avatar
2 votes
1 answer
652 views

Extension of outer unit normal vector to interior

Suppose we have a bounded smooth domain $\Omega$ in $\mathbb{R}^n$, so there exists an outer unit normal vector field $\eta$ everywhere on the boundary. Can we extend it to the interior satisfying ...
H-H's user avatar
  • 237
2 votes
2 answers
446 views

Reconciling some result about the exponential map, the Chow-Rashevskii theorem, and $\mathrm{Diff}_0(M)$

Let $M$ be a $C^{\infty}$ manifold $C^{\infty}$-diffeomorphic to $\mathbb{R}^d$. I've recently come across some results which I'm trying to reconcile. Let $\mathfrak{X}(M)$ denote the set of ...
ABIM's user avatar
  • 5,405
3 votes
0 answers
58 views

Criteria for density of subgroup of diffeomorphism group

Let $C^{\infty,\star}(\mathbb{R}^d)$ denote the non-commutative topological group of smooth diffeomorphisms from $\mathbb{R}^d$ to itself with $\circ$ as multiplication and let $\emptyset\subset X\...
ABIM's user avatar
  • 5,405
5 votes
3 answers
2k views

Functional derivatives on Banach spaces

Physicists often use functional integrals and I'm trying to make sense of it in more precise terms. As you can see here, the functional derivative in Physics is defined in terms of Taylor expansions. ...
JustWannaKnow's user avatar
2 votes
1 answer
301 views

Density of continuous functions to interior in set of all continuous functions

Let $M$ be an $m$-dimensional manifold and $N$ be an $n$-dimensional manifold with boundary. Suppose also that the topology on $N$ can be described by a metric. Thus, the set $C(M,N)$ can be endowed ...
ABIM's user avatar
  • 5,405
2 votes
0 answers
58 views

Is keeping the kernel fixed an open condition for maps of vector bundles?

More precisely, let $M$ be a smooth manifold, $E_1$, $E_2$ vector bundles over $M$, and consider a $C^\infty(M)$-linear map $A:\Gamma(E_1) \to \Gamma(E_2)$ of vector bundles. Now consider the ...
user337331's user avatar
7 votes
0 answers
237 views

Understanding the odd-dimensional index

Given a Dirac operator $D$ on a closed odd-dimensional manifold $M$, I've sometimes heard it said that the Fredholm index of $D$ vanishes because it is an ungraded self-adjoint operator, so that $\dim\...
geometricK's user avatar
  • 1,903
5 votes
0 answers
191 views

Index of the Fredholm operator

I have two vector bundles $E_1$, $E_2$ over $M$ and an embedding of the smooth sections $\lambda : \Gamma(M, E_1) \rightarrow \Gamma(M, E_1 \oplus E_2)$. I consider a Fredholm differential operator $...
Aleksandr Alekseev's user avatar
13 votes
2 answers
655 views

Random matrix with given singular values

Let $\sigma_1\geq\sigma_2\geq...\geq\sigma_n\geq0$ be any deterministic sequence of positive real numbers such that $\sum_{i=1}^n\sigma_i^2=1$. Let $$D=diag\{\sigma_1,...,\sigma_n\}\in\mathbb{R}^{n\...
neverevernever's user avatar
25 votes
2 answers
2k views

Functional approach vs jet approach to Lagrangian field theory

Context: I am a PhD student in theoretical physics with higher-than-average education on differential geometry. I am trying to understand Lagrangian and Hamiltonian field theories and related concepts ...
Bence Racskó's user avatar
2 votes
2 answers
200 views

MP critical point has morse index 1, proof

I wonder where can I find a proof of the following fact: if the mountain pass critical point is non-degenerate, then its Morse index is 1. I am very interested in reading it. In general, I am ...
R. N. Marley's user avatar
0 votes
0 answers
97 views

Smooth sections of finite dimensional bundle and covering space

Let $G$ be a discrete finitely generated group which acts properly and freely on a smooth manifold $M$ with compact quotient $M/G$. Is it right to consider any function $f \in C^{\infty}_c(M)$ (with ...
Aleksandr Alekseev's user avatar
8 votes
1 answer
496 views

Is $C^{\infty}(M)$ dense in weighted Sobolev space $W_{X}^{1}(M)$?

Let $M$ be a compact manifold without boudary and let $X_{1},\ldots,X_{m}$ be smooth vector fields on $M$. Consider the following weighted Sobolev space: $$ W_{X}^{1}(M)=\{f\in L^{2}(M)|X_{j}f\in L^2(...
pxchg1200's user avatar
  • 287
2 votes
1 answer
843 views

Proof of Helmholtz-Hodge decomposition, poor man's version

Helmholtz (-Hodge) decomposition commonly used in physics includes decomposition of a (sufficiently smooth) vector field $F = -\mathrm{grad}(U) + \mathrm{curl}(W)$ on bounded simply connected domain $\...
Ivica Smolić's user avatar
1 vote
1 answer
151 views

A marginal space splitting $\{ \psi \}^{\perp}$

Let $\psi \in L^2(\mathbb R^2,\mathbb C)$. Is there a continuous projection from $\{ \psi \}^{\perp}$ onto $$ \left\{ \varphi \in L^2(\mathbb R^2) \:\:\Big| \int \overline{\psi}(x,y) \varphi(x,y)\...
Alfred's user avatar
  • 31
3 votes
0 answers
91 views

Pseudodifferential operator associated to a self-adjoint extension of a symmetric operator on an incomplete manifold

Let $D$ be the Dirac operator acting on a spinor bundle $S$ over a complete Riemannian manifold $M$. Then $D$ is an essentially self-adjoint operator on $L^2(S)$. Suppose there is a compact subset $K\...
geometricK's user avatar
  • 1,903
4 votes
1 answer
727 views

Simplicity of the first Laplace-Beltrami eigenvalue on Riemannian manifolds

On a compact Riemannian manifold $M$ (we assume Dirichlet boundary condition if $\partial M \neq \emptyset$), the Laplace-Beltrami operator $-\Delta$ has a discrete spectrum $0 < \lambda_1 \leq \...
user144878's user avatar
0 votes
0 answers
149 views

Compact embedding of the $\mathcal{C}^k$ norm on a compact Kahler manifold

Given a smooth complex valued function $f$ on a Kahler manifold $X$, we can define its $\mathcal{C}^k$ norm to be $\sum_{p+q \leq k, 0 \leq p \leq q} sup_{X}|\nabla^{p} \overline{\nabla^q} f|_g$, ...
archer's user avatar
  • 1
5 votes
1 answer
743 views

Eigenvalues and Domain of the Laplace-Beltrami Operator

Assume $(M,g)$ is a compact Riemannian manifold without boundary, where $g$ is the Riemannian metric. Let $L:=-\Delta$ be the Laplace-Beltrami operator on $M$ defined by $\Delta \cdot = \text{div}(\...
MartinG's user avatar
  • 51

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