All Questions
Tagged with differential-operators dg.differential-geometry
171 questions
2
votes
1
answer
123
views
Keeping track of limit cycles via certain second order differential operator
Inspired by the two posts which are linked bellow we ask the following question:
Question: For a vector field $X$ on the plane we define the differential operator $D$ on $C^{\infty}(\mathbb{R}^2)$ ...
- 356
58
votes
22
answers
12k
views
Which high-degree derivatives play an essential role?
Q. Which high-degree derivatives play an essential role
in applications, or in theorems?
Of course the 1st derivative of distance w.r.t. time (velocity), the 2nd derivative (acceleration),
and the ...
0
votes
0
answers
62
views
A uniform upper bound for Fredholm index of Laplace quasi-operators on a compact parallelizable manifold
Assume that $M$ is a compact parallelizable manifold. Is there an upper bound for the absolute value of Fredholm index of all operators in the form $D=\sum_{i=1}^n \partial^2/\partial{X_i^2}$...
- 356
0
votes
0
answers
77
views
A kind of heat equation on a foliated 3D manifold whose leaves are invariant under the flow of a vector field
Assume that $\mathcal{F}$ is a foliation of a $3$ dimensional compact Riemannian manifold $M$ which is invariant under flow of a transversal non vanishing vector field $X$. Let $\Delta_{\...
- 356
4
votes
0
answers
215
views
Exterior derivative on loop space
Notations:
Let $X$ be a manifold, and denote by $LX := C^\infty(S^1,X)$ its loop space. For a loop $\gamma \in LX$ we can think at the tangent space of $LX$ at the point $\gamma$ as the space of ...
- 562
18
votes
2
answers
2k
views
Eigenvalues of the Laplace-Beltrami operator on a compact Riemannnian manifold
Let $(M,g)$ be a compact Riemannian manifold, and let $\Delta_g$ be its Laplace-Beltrami operator. A "well-known fact" is that the eigenvalues of $\Delta_g$ have finite multiplicity and tend to ...
- 993
5
votes
1
answer
457
views
An alternative representation of the principal symbol of the Laplace operator
Assume that $(M,g)$ is a $n$ dimensional Riemannian manifold. We denote by $\Delta$, the Laplacian associated to this Riemannian structure.
Are the following two conditions equivalent?
First condition ...
- 356
0
votes
2
answers
289
views
Derivations of $\chi^{\infty}(M)$ which are elliptic operator
What is an example of a manifold $M$ with $\dim(M)>1$ whose Lie algebra $\chi^{\infty}(M)$ of smooth vector fields admit an elliptic operator $D:\chi^{\infty}(M)\to \chi^{\infty}(M)$ such ...
- 356
5
votes
0
answers
218
views
A differential operator analogy of certain fact in real analysis of smooth functions
Let $E\to M$ be a smooth vector bundle over a smooth manifold $M$.
Let $D$ be a differential operator defined on the space $\Gamma(E)$ of smooth sections of $E$. We fix a section $s\in \Gamma(E)$.
...
- 356
3
votes
0
answers
53
views
Controlling a Schwartz kernel near the diagonal
Let $D$ be a first-order elliptic differential operator that is essentially self-adjoint on $L^2(\mathbb{R}^n)$. Consider the operator $(D+i)^q$ acting on $L^2(\mathbb{R}^n)$ with domain $C_c^\infty(\...
- 1,903
1
vote
0
answers
76
views
PDE on an open ball with prescribed value on some open subsets
Suppose that we have a $r$-order differential operator $L:C^r(B^n, \mathbb{R})\to C^0(B^n,\mathbb{R})$ where $B^n $ is the open unitary ball in $\mathbb{R}^n$ (we can assume for simplicity that it ...
- 2,533
2
votes
1
answer
144
views
Perturbation of the adiabatic limit
Let $(M,g_M)$ be a closed oriented Riemannian manifold that has a fibration structure
$$
Y \rightarrow M \overset{\pi}{\rightarrow} B
$$
where $(Y,g_Y)$ and $(B,g_B)$ are closed Riemannian manifolds ...
- 2,535
2
votes
0
answers
91
views
A quantity associated to a compact Riemannian manifold with boundary(The pair of Laplacian)
Let $M$ be a Riemannian manifold with boundary $\partial M$.
Are the following operators, Fredholm operators?Is there a geometric terminology and geometric interpretation for the fredholm index of ...
- 356
2
votes
0
answers
95
views
Vector bundle endomorphism diffeomorphism invariant?
Let's say we have a vector bundle $V$ on a compact Riemannian manifold $M$ (of dimension $m$, with metric $g$). Given a differential operator $P$, acting on the sections of $V$, of the form: $$P = \...
- 91
5
votes
0
answers
273
views
Is there any geometrical/homological intuition behind symmetrized gradient?
The gradient/differential/exterior differential/divergence/curl are all strictly related first order differential operators. As far as I understood, they are the base of (co)homological theories in ...
- 980
5
votes
0
answers
140
views
Family of Hodge decomposition
It is known that a metric $g$ gives a Hodge decomposition:
$$
\Omega^*(M)=\mathcal H^*(M)\oplus d\Omega^*(M) \oplus \delta_g \Omega^*(M)
$$
Note that the usual differential restricts to an isomorphism ...
- 2,789
3
votes
0
answers
81
views
Conformal manifolds produce Fredholm modules-pseudodifferential operator
This question is a continuation of the discussion which can be found here. From the exterior derivative one constructs an operator $S$ with the property that the graph of $S$ is the (closure of) the ...
- 9,330
14
votes
1
answer
577
views
Projective-invariant differential operator
This question was originally asked on Math StackExchange.
Suppose we want a differential operator $T$ acting on functions $\mathbb{R}^n \rightarrow \mathbb{R}^n$ such that
\begin{align*}
&T(g) = ...
- 2,203
3
votes
0
answers
86
views
Some questions on defining the analytic index
The questions I have are about the definition of the analytic index of a family of self-adjoint Fredholm operators parameterized by a compact space $B$ (say a closed manifold). Actually, the ...
- 1,173
2
votes
0
answers
197
views
Existence of a certain kind of compact spin manifold with boundary
For a compact spin Riemannian manifold $(M^n,g)$ without boundary, $n \not\equiv 3\mod 4$, it is well-known that the Dirac operator associated with a fixed spin structure $S\rightarrow M$ has real, ...
- 121
4
votes
1
answer
331
views
Existence of solution for the PDE $p \frac{\partial f(p, q)}{\partial p}-q \frac{\partial f(p, q)}{\partial q}=g(p, q)$
Consider the following PDE:
\begin{equation}
p \frac{\partial f(p, q)}{\partial p}-q \frac{\partial f(p, q)}{\partial q}=g(p, q),\tag{$\star$}
\end{equation}
where $g$ is a flat function at the point (...
- 307
3
votes
1
answer
373
views
Asymptotic expansion of heat operator $e^{-\Delta{t}}$ and $e^{-\mathcal{D}t}$ of Dirac operator
For a closed Riemannian manifold $M$ of $n$-dimension, we consider the Laplace-Beltrami operator $\Delta$.
It is known that we have an asymptotic expansion for the trace of heat operator $e^{-\Delta{...
3
votes
0
answers
129
views
Differential operators on a compact Lie group associated to bracket-generating sets
Let $G$ be a compact connected Lie group of dimension $m$ with Lie algebra $\mathfrak g$.
Let $\{X_1,\dots,X_h\}$ be a linearly independent set of $\mathfrak g$.
Assume that $\{X_1,\dots,X_h\}$ is ...
- 2,446
7
votes
1
answer
332
views
Atiyah-Patodi-Singer for manifolds with cusps
Dear Colleagues and Friends,
Please let me know if you are aware of any references to the following question.
The classical result of Atiyah, Patodi and Singer tells us that if $W$ is a compact ...
- 1,268
1
vote
0
answers
59
views
Ellipticity of certain differential operator associated to a pair of vector field via curvature tensor
What is a precise example of the following situation:
A compact Riemanian manifold $M$ admits two vector field $X,Y$ such that the the operator $$Z\mapsto R(X,Y)Z$$
Would be an elliptic operator and ...
- 356
7
votes
1
answer
601
views
Universal enveloping algebra and the algebra of invariant differential operators
Let $G$ be a Lie group and $\mathfrak{g}$ be its Lie algebra. Then $\mathfrak{g}$ may be interpreted as the Lie algebra of right (equivalently left) invariant vector fields. Let $\mathcal{U}(\mathfrak{...
- 9,330
5
votes
1
answer
186
views
Reference for Weyl's law for higher order operators on closed Riemannian manifolds
I am looking at page 32 (beginning of Chapter 5) here. We are given a formally self-adjoint, metrically defined differential operator $A$ on $(M^n,g)$ of order $2l$ with positive definite leading ...
- 153
3
votes
0
answers
112
views
Is the square root of curl^2-1/2 a natural (Dirac-)operator?
In current computations on a particular $3$-dimensional Riemannian manifold, a first order differential operator $D:\Gamma^\infty(TM,M)\to \Gamma^\infty(TM,M)$ acting on vector fiels shows up, with ...
- 1,942
5
votes
2
answers
304
views
Simplification of integral on the sphere
In the article: https://arxiv.org/abs/0906.3217 the authors prove in Lemma 1 a formula which helps compute more easily the integral of the Hessian of a function defined on $\Bbb{S}^2$. More precisely, ...
- 2,222
8
votes
0
answers
483
views
Measuring the non-commutativity of the codifferential and pullbacks
$\newcommand{\id}{\operatorname{Id}}$
$\newcommand{\TM}{\operatorname{TM}}$
$\newcommand{\Hom}{\operatorname{Hom}}$
$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\R}{\mathbb{R}}$
$\newcommand{\N}{\...
- 6,741
1
vote
1
answer
126
views
Invariance of the space of harmonic functions under derivation associated to a non-vanishing vector field
Let $X$ be a non-vanishing real analytic vector field on an open manifold $M$. What kind of obstructions would appear when we search for a Riemannian metric on $M$ such that the space of ...
- 356
1
vote
1
answer
157
views
The index of certain differential operator on tori
Assume that $J$ is an almost complex structure on torus $\mathbb{T}^2$. Let $X$ be a non vanishing vector field on the torus. Let $g$ be a Riemannian metric with corresponding $LC$ connection $...
- 356
2
votes
1
answer
155
views
Realization of symbol of Laplace operator via certain integral
Is there an elliptic operator $D$ on $C^{\infty}(S^2)$ whose principal symbol is not identical to thats of Laplacian but it satisfies $\int_{S^2} fDf =\int_{S^2} f\Delta (f)$ for all $f\in C^{\infty}(...
- 356
7
votes
1
answer
535
views
Diffeomorphisms on a real manifold whose derivative are holomorphic maps on the tangent bundle
Edit: According to the answers to the linked MSE question and the comment of Holonomia, I understand that the answer to the second question is that " Every tangent bundles is a complex ...
- 356
8
votes
1
answer
825
views
Semantics of derivations as derivatives
My understanding of how derivations on commutative rings are like derivatives is that a derivation on $R$ is differentiation with respect to a vector field on $\text{Spec}(R)$. But derivations are ...
- 2,130
10
votes
1
answer
848
views
Boundary terms of formal adjoints of differential operators
Let $M$ be a compact manifold with boundary. If we have two vector bundles $E, F \to M$ with inner products and a differential operator $D: C^{\infty}(E) \to C^{\infty}(F)$ then $D$ admits a formal ...
- 1,601
1
vote
1
answer
161
views
Continuity of image of resolvent operator with respect to resolvent parameter
Suppose $D$ is a first-order differential operator on a manifold $M$ and that the inverse $(D+t)^{-1}:H^0(M)\rightarrow H^1(M)$ exists for all $t > 0$, where $H^i(M)$ is the $i^\text{th}$ Sobolev ...
- 1,903
4
votes
0
answers
134
views
Spin equivariance of the Dirac operator-flat case
This question was posed on Math.SE but no one has answered it; it may be suitable for MathOverflow.
Let $D$ be the Dirac operator on $\mathbb{R}^n$ i.e. $D=\sum_{j=1}^nE_j\frac{\partial}{\partial ...
- 9,330
28
votes
6
answers
3k
views
Why is there no symplectic version of spectral geometry?
First, recall that on a Riemannian manifold $(M,g)$ the Laplace-Beltrami operator $\Delta_g:C^\infty(M)\to C^\infty(M)$ is defined as
$$
\Delta_g=\mathrm{div}_g\circ\mathrm{grad}_g,
$$
where the ...
- 1,942
6
votes
0
answers
116
views
Topological constraints for existing of certain differential operators on manifolds
At the beginning a word of warning: this would be rather vague question: vague as it is, I'm not requiring a precise answer, rather some intuitive explanation.
In the flat case $M=\mathbb{R}^n$ ...
- 9,330
3
votes
1
answer
356
views
A second cohomology class associated to a Riemannian manifold
Let $(M,g)$ be a Riemannian manifold of dimension at least $4$.
We consider the differential operator $$D:\Gamma(TM)\to \Gamma (TM)$$
with $$D(X)=\nabla \circ Div(X)$$.
The principal ...
- 356
1
vote
1
answer
94
views
a question about complex Hessians on complex tori
Suppose we have a real-valued smooth function on a complex torus:
$$f: \mathbb{C}^n/(\mathbb{Z}+\sqrt{-1}\mathbb{Z})^n\longrightarrow\mathbb{R},$$
i.e., this $f$ is a real-valued smooth function on $\...
- 593
4
votes
1
answer
525
views
Alternative definitions of infinite-order differential operators
Given (not-necessarily linear) fibre bundles $E \to M$ and $F \to M$ over a smooth manifold $M$ with structure sheaf denoted $\mathcal{O}_M$, a (not-necessarily-linear) differential operator taking ...
- 6,025
1
vote
0
answers
308
views
A differential operator associated with a vector field on the torus
Assume that $X$ is a non vanishing vector field on the torus $\mathbb{T}^2$.
We define two linear operators $T,S$ on the space of smooth functions on $\mathbb{T}^2$ as follows:
$T(f)=...
- 356
4
votes
0
answers
217
views
Self-Dual Equations Elliptic Complex
I'm posting this on MathOverflow because I'm not sure if it would get much of a response on Math.SE. Please feel free to remove it if it is not "research-level".
I was studying Donaldson's self-dual ...
- 1,601
6
votes
1
answer
1k
views
Laplace-Beltrami and the isometry group
H$\vphantom{a}$i. Consider the Laplacian on $\mathbb R^n$,
$$
\Delta=\partial_i^2
$$
It is easy to prove that the most general differential operator that commutes with rotations and translations is ...
6
votes
2
answers
1k
views
The adjoint operators as elliptic operators
Edit:
It seems that the link "https://cms.math.ca/Events/Toulouse2004/abs/ss7.html#lt" which contains a talk by Loic Teyssier about homological equations and vanishing cycles is temporally ...
- 356
1
vote
1
answer
98
views
Conjugacy of $L_X$ operators
Assume that $X$ is a vector field on a $n$ dimensional manifold $M$.Let $0\leq i,j \leq n$.
1.Is there a linear isomorphism $T:\Omega^i(M) \to \Omega^j(M)$ with $L_X T=T L_X$?
2.Is there a linear ...
- 356
2
votes
0
answers
188
views
Singular integrals on compact manifolds
I've been trying to find a reference for a certain thing for a while now, without success. My setting is the following.
I've a compact smooth Riemannian manifold $M$. For $r>0$ (which is supposed ...
- 237
8
votes
0
answers
588
views
Can we use sheaf cohomology to say anything interesting for vector bundles with non-flat connections?
Given a vector bundle $E \to M$ with connection $\nabla$, we get a twisted de Rham sequence using the exterior covariant derivative:
$$0 \to \mathcal{E} \xrightarrow{d^\nabla} \Omega^1_M \otimes \...
- 6,025