All Questions
Tagged with reference-request dg.differential-geometry
800 questions
2
votes
0
answers
195
views
Reference for connection of a Hessian metric
Let $(M,\langle \cdot,\cdot\rangle)$ be a pseudo-Riemannian manifold and $f: M \to \Bbb R$ be a smooth function. One can consider the covariant Hessian $\nabla ({\rm d}f)$. Some time ago I had seen a ...
- 1,163
2
votes
0
answers
75
views
Notation and geometry facts in a paper on the Diederich-Fornæss index
I am reading this article by Bingyuan Liu on the Diederich-Fornæss index.
I am having some problems with both the notation and the geometrical side.
1)I don't know what kind of objects $N,L$ are ...
- 779
2
votes
0
answers
66
views
A boundary for integrals of eigenfunctions over geodesics?
Let $X$ be a compact hyperbolic surface, and $\gamma$ a closed geodesic on it.
Consider the integral
$$\int_\gamma f(x)\, dl(x)$$
where $f$ is a (normalized) Laplace eigenfunction on $X$. ...
- 6,891
2
votes
0
answers
133
views
Variational computations using the moving frame
I'm attempting to learn how to do variational calculus using the method of moving frames, similar to Robert Bryant's answer found here:
Variation of curvature with respect to immersion?
To that end, ...
- 161
2
votes
0
answers
195
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Ampleness under finite map
Let $f:X \rightarrow Y$ be a finite surjective map between quasi-projective varieties. Let L be a line bundle on Y. Suppose $f^*L$ is ample on X. Is it true that L is ample on Y?
How about converse?
user111251
2
votes
0
answers
305
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"Riemannian" collar theorem
Let $(M,g)$ be a compact manifold of dimension $n$ with boundary. If $\partial M$ is smooth then one has a control on the determinant of the Jacobian of the diffeomorphism in the collar theorem, i.e....
- 143
2
votes
1
answer
287
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Regularity of the reparametrization map between curves [closed]
I am looking for a reference for the following kind of results.
Let $\Gamma$ be the space of Lipschitz curves $\text{Lip}([0,1]; \mathbb R^d)$ equipped with the sup norm.
Let $B$ be a Borel subset of ...
- 980
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
2
votes
0
answers
346
views
What information is encoded in discriminant varieties?
I'm looking for Web-accessible references that survey the connnections among the following constructs:
discriminant varieties
Vandermonde matrix/determinant/polynomial
moment curves for the n-...
- 10.5k
2
votes
0
answers
127
views
Functional inequality under mean curvature flow
Let $\Sigma$ be a hypersurface in $\mathbb R^n$ and $\Sigma_t$ be a variation of $\Sigma$ under the mean curvature flow under an extra condition that ${\rm vol}_{n-1}(\Sigma)={\rm vol}_{n-1}(\Sigma_t)$...
- 143
2
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0
answers
138
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Geometric flow of the total tension functional
I apologize if this question is silly or confusing, I am completely new to this subject.
Let $(M,g)$ be a Riemannian manifold. Denote by $\nabla$ the Levi-Civita connection of $(M,g)$. Now, let $S^{n}...
- 395
2
votes
0
answers
145
views
Semistability of a sheaf on nodal curve
Suppose $X$ is a projective, connected, nodal curve (can be reducible) over an algebraically closed field $k$ of arbitrary characteristic. Let $F$ be a pure sheaf on $X$ and denote by $\pi^{*}(F)$ its ...
- 2,323
2
votes
0
answers
67
views
On two functions with isodirectional gradients
Let $U\subset \mathbb{R}^n$ be open and $f,g:U \to \mathbb{R}$ be two $C^1$ functions whose gradients are always in the same direction, i.e. $\forall i,j \in \left\{1,...,n\right\}$
\begin{equation}
(\...
- 1,461
2
votes
0
answers
80
views
Largest disk inside a spherical domain
It is known (Pestov-Ionin theorem) that if $k_{max}$ is the maximum curvature of a smooth planar loop $\gamma$, then there is a disk of radius $1/k_{max}$ inside $\gamma$. I wonder is there any ...
- 175
2
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0
answers
232
views
Intuitive understanding of the mean curvature flow [closed]
I am trying to develop some intuition into the properties of mean curvature flow of a surface in $\mathbb{R}^3$. As an example, I am trying to understand what happens to a surface of revolution $S = (...
- 21
2
votes
0
answers
344
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If the fibers of a submersion are connected, does it mean that any 2 sections are homotopic (locally on the base)?
Is the following fact known? If yes - what is the reference?
Let $\phi:X\to Y$ be a submersion of smooth manifolds with connected fibers.
Let $s_0,s_1:Y\to X$ be its (smooth) sections. Then, for any $...
- 2,649
2
votes
0
answers
234
views
Concentration compactness on a compact setting
Consider a compact Riemannian manifold $M$ of dimension $n$ and a sequence of positive functions $\{\varphi_k\}_k \in C^\infty(M)$ such that $\{\varphi_k\}_k$ satisfy the basic concentration ...
- 1,407
2
votes
0
answers
187
views
Heat kernel on manifold with boundary
Let $M$ be a Riemannian manifold with boundary, let $\mathcal{V}$ be a metric vector bundle over $M$ and let $L$ be a formally self-adjoint Laplace type operator, acting on sections of $\mathcal{V}$. ...
- 9,853
2
votes
0
answers
123
views
Mean value operator on Riemannian manifold
Let $(M,g)$ a Riemannian manifold. Further $M$ should be a harmonic space, that is $M$ is a symmetric and simply connected space of rank 1. (Example: Spheres $S^n$)
Consider the mean value operator, ...
- 21
2
votes
0
answers
58
views
Mathematical Billiards: Set of starting points and velocities hitting boundary at time t
In the simplest setting $\Omega$ smooth compact and convex in $R^n$ with linear constant speed trajectories that is ($q_t=q_0+t\cdot v$ until the collision point). What is known about the structure ...
- 1,256
2
votes
0
answers
59
views
Group of real analytic isometries of $g$-fold product of the Poincare upper half plane
Let $\mathfrak{h}^g$ be the cartesian product of $g$ copies of the Poincare upper half plane. We endow $\mathfrak{h}^g$ with the usual Poincare metric given in local coordinates by $ds^2=\sum_{i=1}^g ...
- 7,609
2
votes
0
answers
108
views
Quantitative estimate of heat dispersion - off diagonal estimates
Consider the heat equation $\partial_t u - \Delta u = 0$ on a compact manifold $M$ (if $M$ has a smooth boundary, then we assume either Dirichlet or Neumann boundary condition). Consider $u_0 (x) = u(...
- 21
2
votes
0
answers
168
views
Sources on evolution of submanifolds subject to Ricci flow
I am seeking any textbook or paper addressing the evolution of submanifolds of a manifold undergoing Ricci Flow. Please, any pointer towards this topic is more than welcome.
This old MO post may be ...
- 163
2
votes
0
answers
179
views
Questions about transformation or integral transformation
I have asked several mathematicians about the following questions,but all of them think they are good questions,but can not give a complete answer.Now I have to come here to ask mathematicians all ...
- 3,094
2
votes
0
answers
282
views
Generalizing a result of Paul Andi Nagy
I am trying to generalizate a result of Paul Andi Nagy which says that in a almost Kähler manifold with parallel torsion we have $\langle \rho,\Phi-\Psi\rangle $ is a nonnegative number; in fact
$$
4\...
- 21
2
votes
0
answers
189
views
About the Lie algebra of polyvector fields
I would like to know if someone already did some computations of the group of Lie algebra automorphisms of the algebra of polyvector fields on $\mathbb{R}^n$ equiped with the Schouten bracket (or ...
- 1,609
2
votes
0
answers
260
views
Perturbation of Morse function at a critical point
I recently learned from a knowledgeable person that for a Morse function $f: M \to R$ with a critical point $x_0$, one can perturb $f$ in such a fashion that the new function has the same critical ...
- 1,211
2
votes
0
answers
115
views
Special class of bi-hamiltonian systems
A bi-Hamiltonian manifold is a manifold $M$ equipped with two compatible Poisson tensors $\pi_0$ and $\pi$.
I am interested in the case of a Lie group $G$ endowed with a multiplicatif Poisson tensor $...
- 513
2
votes
0
answers
536
views
Space of derivations of holomorphic (analytic) functions
Let $M$ be a (real) smooth manifold, and $p \in M$.
The space of (linear) derivations $D:C^{\infty}(p) \to \mathbb{R}$ (i.e., maps satisfying $D(f+g) = D(f)+ D(g)$ and $D(fg)=D(f)g(p) + f(p)D(g)$) ...
- 1,058
1
vote
2
answers
314
views
Notion of manifold curvature?
Consider a particular embedding of a $C^2$ manifold $\mathcal{M}\subseteq\mathbb{R}^m$. Given $p\in\mathcal{M}$, suppose $\epsilon>0$ is small enough that the portion $U$ of $\mathcal{M}$ which is ...
- 7,633
1
vote
3
answers
502
views
orbit space of $\mathbb{Z}_p$ action over complex projective space by permuting the homogeneous coordinates
$Z_p$:=cyclic group of order $p$.
I want to understnd $H_\ast(\mathbb{C} P^{n}/Z_{n+1};Z)$ with $(n+1)$ being a prime number,and the action is given by permuting the homogeneous coordinates.
For ...
- 593
1
vote
1
answer
232
views
The bundle of symmetric affine connections as quotient of the second-order frame bundle
This post is not about finding an answer to a certain problem - because the answer already exists - but rather about finding the simplest possible answer.
The problem is: how to define the bundle $C(...
- 2,527
1
vote
1
answer
291
views
What is the status of the smooth version of bellows conjecture
Bellows conjecture for polyhedra was setteled in 1997. How about the smooth version of it, ie bending of closed 2D submanifolds in $\mathbb{R}^3$ while preserving the Riemannian structure/intrinsic ...
- 1,117
1
vote
1
answer
265
views
Boundary components of a subsurface
Consider the following situation. Suppose we have a closed oriented Riemannian surface $ \Sigma $ and a connected open subset $ \Omega \subseteq \Sigma $ with a boundary, consisting of finitely many ...
- 337
1
vote
1
answer
243
views
Reference for non-parallel harmonic $k$-forms
I want to get some deep understanding on closed orientable Riemannian manifolds admitting $k$-forms ($k\geq 2$) $\omega$ that satisfices the following conditions:
$$\nabla \omega\neq 0,\quad \Delta\...
- 4,195
1
vote
1
answer
840
views
Reference request: Gauge theory [closed]
What are some good introductory texts to gauge theory? I have some basic differential geometry knowledge, but I don’t know any algebraic geometry.
Also, as a side question, what intuitively is a ...
- 2,069
1
vote
2
answers
1k
views
Reference on Complex Geometry
For the preparation of a complex geometry lecture I am looking for a good literature. I already have standard literature like Huybrechts "Complex Geometry. An Introduction" and I am also using it. But ...
- 39
1
vote
2
answers
534
views
Are there some websites for self learining of advanced mathematics? [closed]
Are there some websites for self learining of advanced mathematics?
For example, some great lecture vedio of differential geometry, Lie group , Lie algebra, algebraic topology and so on. Thanks
- 977
1
vote
1
answer
158
views
Comparison of special metrics on Riemann surfaces with the hyperbolic one
Let $X$ be a complex algebraic curve, or equivalently a compact Riemann surface, of genus $a\ge2$. Denote $\omega_1, \dots , \omega_a$ a basis of holomorphic differentials, and consider the Riemaniann ...
- 31
1
vote
1
answer
255
views
A different version of Besicovitch Covering Theorem involving balls of half radius
I am trying a find a reference to/proof of the following result:
Let $(M, g)$ be a compact Riemannian manifold. Then there is $b$ so that the following holds: for any $r>0$, there is a covering $\...
- 534
1
vote
1
answer
64
views
Unbounded convex domains in 2D
Let $\gamma$ be a smooth planar curve. Assume that $\gamma$ divides the plane into two domains and, it addition, that one of these domains is unbounded and convex. What can be said about the behavior ...
- 175
1
vote
1
answer
208
views
Is there an algebraic way to characterise the ordinary integral flags?
Fix a vector space $V$ and an integer $1\leq n<\dim V$.
If $\mathcal{I}\subseteq\Lambda^\bullet V^*$ is an ideal, I denote by $\mathcal{I}^i:=\mathcal{I}\cap\Lambda^iV^*$ its $i^\textrm{th}$ ...
- 2,527
1
vote
1
answer
409
views
Heat kernel upper bound on compact Riemannian manifold
Let $M$ be a compact Riemannian manifold without a boundary. Let $p_t(x,y)$ be the heat kernel. I am looking for a reference for the result: there exists a constant $C$ such that
$$|p_t(x,y)| \leq C$$
...
- 65
1
vote
2
answers
188
views
Calculating Exterior Distance from Measurements of Inner Geometry
Gauss has proven in his famous Theorema Egregium, that it is possible, to calculate the gaussian curvature from measuring angles and distances on the surface, irrespective of how the surface is ...
- 13.2k
1
vote
2
answers
341
views
Copies of ax+b inside the AN part of an Iwasawa decomposition?
As a relative novice to the structure theory of Lie algebras and Lie groups, the following is what I can gather from reading parts of Helgason's book DG, Lie groups and symmetric spaces and Knapp's ...
- 25.8k
1
vote
1
answer
732
views
Norm equivalent to Sobolev norm? [closed]
On the hyperbolic space $\mathbb{H}^n$, it is known that the spectrum of the Laplacian satisfies $\text{Spec}(-\Delta) \subset [\frac{(n - 1)^2}{4}, \infty)$. Consider the operator $P = -\Delta + a$, ...
- 11
1
vote
1
answer
1k
views
Hermitian form, fundamental $2$-form of Kahler structure on $\mathbb{C}^n$
I've come across the following (it is an excerpt of Stolzenberg's lecture notes 19):
Wirtinger's Inequality.
Let $L$ be a complex linear space and let $M$ be a real
even-dimensional subspace....
- 103
1
vote
2
answers
139
views
Reference request: minimal (maximal) Lorentzian surfaces in $\mathbb{R}^{1,2}$
Let $R^{1,2}$ be the Minkowski 3-space, I would like to know any references about minimal (maximal) orientable Lorentzian surfaces in $\mathbb{R}^{1,2}$, including examples and maybe general theories, ...
- 783
1
vote
2
answers
205
views
Tangent vectors on the algebra of trigonometric polynomials
Let $G$ be a compact real Lie group and ${\sf Trig}(G)$ the algebra of trigonometric polynomials on $G$ (defined in the Hewitt-Ross, Abstract harmonic analysis, (27.7)), i.e. the algebra of functions $...
- 7,426
1
vote
1
answer
574
views
Proof that the Hodge-de Rham Rank Equals the Euler Characteristic
Can someone please provide a good (online accessible) reference for the well-known identity
$$
\text{rank((d + d}^*)^+) = \sum_{i=}^n (-1)^i \dim(H^i(M)),
$$
where $M$ is a manifold of dimension $n$, ...
- 11