All Questions
Tagged with reference-request dg.differential-geometry
800 questions
1
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1
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157
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Commuting time dependent vector fields and pullback invariance
Let $X_t, Y_t \in C^\infty(\mathbb{R}; \mathfrak{X}^\infty(M))$ be (smooth, or something else if it's necessary) time dependent vector fields.
Is there some analogue of the following fact in finite ...
- 193
1
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1
answer
124
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Relative equivariant Thom transversality
I'm looking for a reference for the following:
Suppose that $G$ is a finite group, that $M$ is a smooth $G$-manifold, and that $A\subseteq M$ is a closed $G$-invariant subspace of $M$ such that the ...
- 55
1
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1
answer
258
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Isoperimetric inequality for domains in the exterior of a precompact open set in Riemannian manifold
Fix $n\geq 2$ and let $$\mathbb{H}^{n}=\mathbb{R}_{+}\times \mathbb{S}^{n-1}$$ be the hyperbolic space, so that any point $x\in \mathbb{H}^{n}$ can be represented in polar coordinates $x=(r, \theta)$, ...
- 459
1
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1
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571
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Continuity of the differential flow under a perturbation of the vector field
Suppose $v$ is a (possibly time-dependent) vector field on a compact manifold $M$.
Its flow is a mapping $g: M \times \mathbb{R} \rightarrow M$, where $g$ satisfies the following conditions (written ...
1
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1
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1k
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Approximation of a continuous function by a smooth one on an open set
I'm interested in the following kind of theorems :
Let $M$ be a real analytic manifold and $U$ an open set of $M$. Let $f : U \to \mathbb{R}$ a continuous function. Then, there is a $C_{\infty}$ ...
- 193
1
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1
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211
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Curvature of plane curves on a surface
Let $S$ be a surface and $\gamma$ a curve on $S\subseteq \mathbb{R}^3$ obtained cutting $S$ with a plane. I wuold an upper bound for the curvature of $\gamma$. Are there papers for this topic?
- 1,252
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1
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145
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Continuity of Busemann-Hausdorff area density
I am trying to find out why the Busemann-Hausdorff area density as defined by Burago and Ivanov is continuous. Here, $GC_m(V)\subset \Lambda^m(V)$ denotes the simple $m$-vectors in an $n$-dimensional ...
- 193
1
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1
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252
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Sobolev multiplication $\otimes$ of $H^1=W^{1,2}$ in vector bundles
Let $E\to X$ be a vector bundle with an inner product and fix a reference connection $A_0$ on $E$. Then for $1\leq p < \infty$ and $k\geq 0$ we can define the Sobolev space $W^{k,p}(E)$ as the ...
- 729
1
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1
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361
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Control of the $C^1$ norm of a diffeomorphism
Let $\Omega$ be a smooth open set of $\mathbb{R}^3$ diffeomorphic to the unit ball $B$. Let assumme that the boundary $\partial \Omega=\Sigma$ is also smooth and satisfies:
$$\int_\Sigma H^2 d\sigma \...
- 914
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1
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259
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$L^2$-de-Rham complex on Lipschitz domains has smooth harmonic forms?
I would like to know for which choice of boundary conditions the title statement is true.
Let $\Omega$ be a bounded Lipschitz domain in $\mathbb R^n$, for which we regard the $L^2$-de-Rham complex.
...
- 5,327
1
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1
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280
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Application of Yamabe and Liouville type equation
Let $\Omega$ be a domain in $\mathbb{R}^n$. I am interested in the following critical elliptic partial differential equations (PDEs):
The Yamabe Type Equation (for $n>2$):
\begin{equation}
-\...
- 914
1
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1
answer
149
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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
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1
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94
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Initial value problems on manifolds around submanifolds (reference)
I am looking for a reference on initial value problems formulated on smooth manifolds with initial conditions on submanifolds. More precisely, let $X$ be a smooth manifold and $Y\subset X$ a embedded ...
- 11
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1
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331
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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
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1
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220
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Question on $\alpha-$Einstein manifolds
A Riemannian manifold $(M,g)$ is called $\alpha-$Einstein if there exist a non-zero $1-$form $\alpha$ such
$$\rho=ag+b\alpha\otimes\alpha$$
where $a,b$ are smooth functions on $M$ and $\rho$ is ricci ...
- 4,195
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1
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199
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Relative divisors
Let $X\rightarrow T$ be a fibre bundles with smooth projective fibre $F$ and $X$ and $T$ are also smooth. Let $D$ is relative effective Weil divisor. Suppose $W_1 $ and $W_2$ are relative subvarieties ...
user100841
1
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1
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165
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Can any bounded area defined by polynomial inequality in $\mathbb{R}^n$ be partitioned into simply connected finite area such that
Can any bounded area defined by polynomial inequality in $\mathbb{R}^n$ be partitioned into simply connected finite areas such that for each simply finite area there exist a diffeomorphic map that ...
- 3,094
1
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1
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178
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Casimir of a three dimensional solvable lie algebra
Good morning everyone. I've encountered recently during my computations the following lie algebra
$$\mathfrak g=\text{span}(f_0,f_1,f_2),$$
with $$\begin{eqnarray}[f_2,f_1]&=&f_0+a f_2,\\ [...
user17697
1
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1
answer
117
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Minimal Legendrian submanifolds and laplacian of particular functions
I'm reading the paper
Lê, Hôngvân(D-MPI-NS); Wang, Guofang(PRC-ASBJ-MSY)
A characterization of minimal Legendrian submanifolds in $S^{2n+1}$. Compositio Math. 129 (2001), no. 1, 87–93.
Let $x: L^n \...
- 585
1
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1
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755
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Tensor analysis/Differential forms outside physics
There are many "geometric systems" like tensor analysis or differential forms calculus, which more or less different perspectives onto the same abstract relations.
Most applications are physical, ...
1
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0
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69
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Unique continuation of Laplace eigenforms
Let $M$ be a compact Riemannian manifold and $\Delta = d\delta + \delta d$ denote the (positive definite) Hodge Laplacian acting on differential forms. Call a smooth differential form $\omega$ a ...
- 1,407
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0
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99
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Shortest loop through vertices of a convex polytope
Let $P$ be a convex polytope in Euclidean space $\mathbf{R}^3$ and $\Gamma$ be a closed curve which passes through all vertices of $P$. How small can the length $L$ of $\Gamma$ be? More specifically, ...
- 7,169
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0
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52
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High order parabolic PDEs on manifolds: Reference request
I recently became interested in parabolic PDEs of order 4 on surfaces. However, I have a very little background in parabolic PDEs. I discovered Lunardi's book (Analytic semigroups and optimal ...
- 363
1
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0
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122
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Bilipschitz constants of exponential map on small ball for Riemannian manifold with curvature bounds
Let $(M,g)$ be a Riemannian manifold with sectional curvature $\mathrm{sect}$ between $-K\le \mathrm{sect} \le K$ for some $K>0$. In [1] it is stated at the beginning of section 4, that if $u,v\in ...
- 111
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0
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113
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Hodge-Helmholtz decomposition for 1-form of strategic game
This question is motivated by the attempt of decomposing (in a direct sum sense) a strategic game into components in the spirit of Hodge decomposition.
Preamble
Combinatorial setting
Candogan et al. (...
- 111
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0
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170
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$L^2$ metric on $\textrm{Diff}(M)$ and geodesics
The paper Geometry of diffeomorphism groups, complete integrability and optimal transport mentions the following:
The group $\textrm{Diff}(M)$ carries a natural $L^2$-metric
$\displaystyle \langle\...
- 305
1
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0
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132
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About the classification of simply connected homogeneous 3-manifolds
I've read somewhere (but cannot locate the source) that the following classification holds: simply connected homogeneous 3-manifolds are either isometric to $S^2 \times \mathbb{R}$ or to a metric Lie ...
- 1,763
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0
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72
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A proof that the analytic index for families is multiplicative
I am looking for a detailed proof of the analytic index for families (of geometrically defined Dirac operators is good enough and assuming the existence of the kernel bundle) is multiplicative. Any ...
- 1,173
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0
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310
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Can we classify two dimensional complex vector bundles over $\mathbb{R}P^2$?
I know it is easy to classify line bundles over $\mathbb{R}P^2$. But do we have a classification of two dimensional complex vector bundles over $\mathbb{R}P^2$?
- 9,019
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0
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46
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Help with a surface of delay differential equations
This question is difficult for me to phrase, as it's very much outside of my mathematical purview. This is a question which intersects directly with my research, but as I work predominantly in ...
- 388
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0
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56
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Smooth approximation in Sobolev spaces for surfaces with boundary
Let $\mathbb{D}$ be the unit disk in $\mathbb{C}$ with closure $\overline{\mathbb{D}}$, and let $\varphi:\partial \mathbb{D}\to \partial \mathbb{D}$ be any continuous homeomorphism. Let $\mu$ be a ...
- 163
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0
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138
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References Request: Bach tensor
Recently I want to study about Bach tensor in detail. From the fundemental definition and properties to conformal invariant. Is there any references to me about Bach tensor? If there are some origins ...
- 11
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0
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145
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Integrability conditions imply existence of potential
I'm looking for a proof of the following well-known theorem:
If $f$ is a continuously differentiable vector field in a simply connected region $G\subset \mathbb{R}^n$ which satisfies the ...
- 2,160
1
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0
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170
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Uniqueness of collar neighborhoods for non-compact boundary case in smooth setting
Let $M$ be a smooth manifold and let $f_0, f_1 \colon [0, 1] \times
\partial M \to M$ be two smooth embeddings that are the identity map
on $\partial M \times\{0\} = \partial M$ . If $\partial M$ is
...
- 265
1
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0
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101
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Actions of finite groups on compact symmetric spaces
I am interested in series of finite subgroups of the classical compact simple Lie groups which have big orbits on compact symmetric spaces and where the double coset space has some nice explicit ...
- 8,597
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0
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213
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Injectivity radius bounds for Riemannian manifolds of low regularity
In their seminal paper, Jeff Cheeger, Mikhail Gromov, and Michael Taylor derivated bounds on the injectivity radius of Riemannian manifolds with bounded sectional curvature of the form:
$
inj(p)\geq r ...
1
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0
answers
328
views
Codifferential of wedge of two 1-forms
Let $\omega,\eta$ be two 1-forms on a manifold $M$. I'm interested in an expression for
$$
\delta(\omega\wedge\eta)
$$
where $\delta$ is the co-differential operator $\Lambda^2(M)\to\Lambda^1(M)$. ...
- 213
1
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0
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153
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Reference request for a paper of Berard-Bergery
I was wondering if anyone could point me to a pdf copy of the following paper by Lionel Berard-Bergery:
"Scalar curvature and isometry group", in Spectra of Riemannian Manifolds, Kaigai ...
- 1,903
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0
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55
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Projection of a real analytic manifold onto subspace is union of real analytic submanifolds
Let $M$ be a compact connected real analytic submanifold of the Euclidean space $\mathbb{R}^{n} \times \mathbb{R}$ and denote by $\pi : \mathbb{R}^{n} \times \mathbb{R} \rightarrow \mathbb{R}^{n}$ the ...
- 19
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0
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77
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Stratification of the space of maps transverse to another given one
If $M,N,S$ are manifolds ($M$ and $S$ compact) and $g:S\to N$ is a smooth map, then if we endow $C^\infty(M,N)$ with the strong topology we get that
$$\pitchfork(M,N;g):=\{f\in C^\infty(M,N)\,;\,f\...
- 31
1
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0
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137
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Invariant subspace of a nonlinear map
First please see this very simple fact:
Fact: $\ $ Any linear map $T: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ has a proper invariant linear subspace.
By an invariant subspace we mean a space $M$ ...
- 111
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0
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61
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Splitting formulas for spectral flows
I'm asking if there are splitting formulas for equivariant spectral flow and higher spectral flow (of Dai-Zhang) for paths of Dirac operators, concerning gluing together two smooth compact Spinc ...
- 11
1
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0
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90
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Is there any name/occurence to this sequence of numbers?
I am curious if there is any name for this sequence of numbers, or any occasion that this sequence is used.
The sequence is $(c_1,c_2,c_3,\cdots)$ with recursive formula
$$c_n=\frac{1}{2n+1}\sum_{i=...
- 323
1
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0
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42
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Effect of plumbing a surface on the marked length spectrum
First I'll recall the plumbing procedure.
Let $M$ be a noded Riemann surface with nodes $p_1,\dots, p_n$. There is a family of pairwise disjoint neighbourhoods of each node $U_i$ that has coordinates ...
- 445
1
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0
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159
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The Laplacian of a tubular neighborhood
Let $(M,g_M)$ be a compact submanifold of $\mathbb{R}^n$. Are there any known results relating the spectrum of the Laplace-Beltrami operator of M to the spectrum of the Laplace-Beltrami operator of a ...
- 409
1
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0
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126
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Paper "Tetrahedron rolled onto a plane"
I am looking for the article Charles W. Trigg, "Tetrahedron rolled onto a plane", J. Recreational Mathematics, 3(2):82–87, 1970.
It is from @Joseph O'Rourke comment in the previous post "Die-rolling ...
- 11
1
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0
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75
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Derivation of the vortex filament equation from Euler equation
How can the vortex filament equation
$$\partial_t \chi = \partial_s \chi \wedge \partial_{ss} \chi,$$
where $\chi(t,s)$ is a curve in $\mathbb R^3$,
be derived from the Euler equation
$$\partial_t \...
- 277
1
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0
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68
views
Curvature of projection function onto a smooth curve
Suppose we have a smooth curve $C$ lying in $\mathbb{R}^2$, and let us consider the orthogonal projection function $P_C(x)$ onto the curve, described by
$$P_C(x) = argmin_{y \in C} \Vert x - y \Vert$$...
- 141
1
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0
answers
96
views
The hermitian Einstein manifolds
I take an hermitian manifold $(M,g,J)$ and I define from the riemannian curvature $R(X,Y)$ as a $2$-form:
$$
Ricc(J)= \sum_i R(J e_i,e_i)
$$
with $(e_i)$ an orthonormal basis of the tangent.
$$
2R(J)=...
- 187
1
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0
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272
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A cohomology associated to a Riemannian manifold
Let $N$ be a compact Riemanian manifold and $G$ be its isometry group. Let $M=\chi^{\infty}(N)$ be the space of smooth vector fields on $N$. There is a natural right action of $G$ on $M$ with $X.g=g^*(...
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