Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.
2
votes
0answers
108 views
undergraduate handle decomposition. Reference
As the title says, I'm searching for a nice textbook for introducing the theory of handle decomposition of manifolds to undergraduate students.
7
votes
2answers
216 views
Necessary condition on Calabi-Yau manfiold to be a hypersurface in a Fano manifold
Let $X$ be a smooth projective Calabi-Yau threefold. Are there any known obstructions to it
being a member of a base-point-free linear system in a nef-Fano fourfold?
What, in anything, is known ...
1
vote
1answer
146 views
Solvable question of dee dee bar lemma
Recently I read about the dee dee bar lemma ($\partial\bar \partial$-lemma) in Gang Tian's Canonical metrics in Kähler Geometry. In the middle of Page 16, the author writes that: "The following ...
1
vote
3answers
178 views
Two questions on isometric embedding
According to the answer of the following question, I try a new version:
An special isometric embedding
Let $M$ be a Riemannian manifold and $\gamma$ be a small part of a geodesic.
Is there an ...
0
votes
1answer
60 views
An special isometric embedding
Let $M$ be a Riemannian manifold and $\gamma$ be a non closed geodesic.
Is there an isometric embedding of $M$ into some $\mathbb{R}^{n}$ which send $\gamma$ into an straight line?
The second ...
3
votes
1answer
166 views
What is the canonical form of real symmetric 2n\times 2n matrix under unitary congruence?
If $M$ is a $2n\times 2n$ real symmetric matrix, I would like to ask what could be its canonical form under unitary congruence. We view a unitary $n\times n$ matrix $U$ as a real $2n\times 2n$ matrix, ...
1
vote
1answer
187 views
A question about complex polarization
Let $M$ be a symplectic manifold, Then a subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if
$P$ is Lagrangian
P involutive
dim$P\cap\bar P ...
4
votes
1answer
302 views
Triple bubble conjecture: Natural candidate?
Is there a standard natural candidate surface for
the shape that encloses three given volumes
in $\mathbb{R}^3$ and has minimal surface area?
I know the planar triple bubble conjecture was ...
0
votes
0answers
31 views
Existence of local frames in Hilbert manifolds
Thank you for looking at my queation.
I'm studying about Hilbert manifolds. I would like to ask you if there exists some concept like a local frame for every Hilbert manifold.
I'm refering to the ...
11
votes
2answers
325 views
How useful/pervasive are differential forms in surface theory?
Every year I teach an introductory class on the differential geometry of surfaces, including numerical aspects (e.g., how to solve PDEs on surfaces). Historically this class has included an ...
8
votes
0answers
104 views
A simple proof that parallelizable oriented closed manifolds are oriented boundary?
So let $M$ be a smooth closed orientable real manifold such that $M$ is parallelizable, i.e., the tangent space $TM$ of $M$ is trivial. From the triviality of $TM$ we get that the Stiefel-Whitney and ...
1
vote
1answer
67 views
Cut locus, conjugate points and smoothness of distance function
I had a couple of related questions on the cut locus, conjugate points and smoothness of distance function. Let $(M,g)$ be a smooth complete Riemannian manifold and $r(x) = d(p,x)$ the distance ...
2
votes
1answer
144 views
almost holomorphic line bundles
Let $(L,\omega_L) \to (M,\omega_M)$ be a symplectic line bundle (symplectic line=real dimension 2) over a symplectic manifold $M$. Each of these objects can be equipped with an almost complex ...
3
votes
0answers
78 views
Newlander-Nirenberg in dimension 2
What is the easiest (and what is the most elementary) way of proving
Newlander-Nirenberg theorem for Riemannian surfaces? I was able to reduce
it to existence of non-trivial harmonic functions ...
0
votes
0answers
77 views
Existence and uniqueness on this semi-linear parabolic PDE [migrated]
I want to know whether the existence and uniqueness of a classical solution can be found about this question:
Find a classical solution $u : [0,T]\times [0,\infty] \rightarrow {\mathbb R}$, such ...
-2
votes
0answers
177 views
A Bundle associated to a submanifold of $\mathbb{R}^{n}-\{0\}$
Edit:
According to the comment of Rayan Budney I revised the question as follows:
Let $M$ be a submanifold of $\mathbb{R}^{n}-\{0\}$. We define a $n-1$ dimensional bundle
\begin{equation}
...
7
votes
1answer
189 views
Extended Deformation Theory (dg-Lie algebra principle in positive characteristic?)
Recently, I looked at articles that make use of Deligne's idea that "in characteristic 0 every deformation problem is governed by a differential graded
Lie algebra" as explained first in ...
0
votes
0answers
26 views
When is Dirichlet solution from disk to Riemannian manifold Holder continuous near the boundary?
$D$ is the two-dimensional unit disk. $X$ is a compact Riemannian manifold.
Let$\phi \in W^{1,2}(D,X)$ and define
$$
W^{1,2}_{\phi}=\{v \in W^{1,2}(D,X):Trace(v)=Trace(\phi)\}
$$
Let $$
...
2
votes
0answers
66 views
Laplacian on manifolds with corners
So far I've studied smooth riemannian manifolds and the Laplace-Beltrami operator associated to it. Therefore I know basic theorems like Stoke's theorem, divergence theorem, green's identities etc. ...
11
votes
3answers
601 views
Sobolev spaces and geometry
This is a very naive question, is there a way to geometrically understand Sobolev spaces without going through analysis and PDE's? To my knowledge, Sobolev spaces where created precisely to study ...
1
vote
1answer
124 views
The set of leaves of the distribution $D$ on coadjoint orbit $O_{\mu}$
Let $G$ be a compact connected Lie group and $O_{\mu}$ be a coadjoint orbit where $\mu\in \mathfrak{g}^*$ and $\mathfrak{g}^*$ is the dual of the Lie algebra of $\mathfrak{g}=\mathrm{Lie}(G)$. Let ...
1
vote
0answers
80 views
Euler class and self-intersection number of a surface in a 4-manifold [duplicate]
In the first two paragraphs of Circle bundles over $CP^1$ and self-intersection number of $CP^1$ embeddings, it is claimed that
For a compact oriented surface $X$ in a 4-dimensional oriented ...
2
votes
1answer
141 views
Chern class of Hopf fibration over elliptic curve
Let $N = \{ (z_0,z_1,z_2) \in S^5 \mid z_0^3+z_1^3+z_2^3 = 0 \}$, where we consider $S^5\subset\mathbb{C}^3$.
The circle $U(1)$ acts on $N$ by
$$e^{i\theta} \cdot (z_0,z_1,z_2) = ...
1
vote
0answers
60 views
Boundary of fibers of submersions
Let $M$ be a smooth manifold with boundary (say of dimension $m$) and let $N$ be a smooth manifold with no boundary (say of dimension $n$ with $m\geq n$). We have the following classical result:
...
2
votes
0answers
51 views
Hessian in local coordinates [migrated]
Let $f\colon M\to \mathbb{R}$ be a smooth function on a manifold $M$ with a critical point $p$. We define its Hessian at $p$ via $H(u, v)=(UVf)(p)$ where $u, v\in T_pM$ and $U$ and $V$ are vector ...
2
votes
2answers
125 views
Does $(M\times M, \omega\times -\omega)$ admit a Lagrangian fibration?
Let $(M,\omega)$ be a manifold endowed with symplectic form. Then the product manifold $M\times M$ with symplectic form $\omega\times -\omega$ is symplectic, and the diagonal submanifold ...
3
votes
0answers
54 views
tangent developable surface in $\mathbb{R}^3$
Let $C$ be a regular curve embedded in $\mathbb{R}^3$ (i.e. a real 1-dimensional manifold embedded in $\mathbb{R}^3$). Let $S$ be the union of its affine tangent lines:
$$S=\bigcup\limits_{p\in ...
1
vote
1answer
102 views
A question about the “size” of the neighbourhoods in which bundles are trivializable
My question is about domains of trivializability of distributions on a smooth compact manifold. Assume that you have a sequence $\{E^k\}_k$ of $C^r$ distributions of rank $n$ which is $C^0$ close to a ...
2
votes
1answer
116 views
Killing constant in Killing spinor equation
This is a simple question, but I can't find explicit discussion in literatures that I can find. Real/imaginary Killing spinor equation
\begin{equation}
\nabla_\mu \psi = \lambda \gamma_\mu \psi
...
6
votes
1answer
132 views
quantitative version of the rigidity of the 2-sphere
I am looking for a quantitaive version of the following theorem:
A compact surface with $K\equiv 1$ is isometric to the round sphere.
Of course I get the Berger, Brendle-Schoen Theorem which insures ...
15
votes
2answers
369 views
Does a small-area sphere in a 3-manifold bound a small ball?
Let $M$ be a closed riemannian 3-manifold. I think that the following fact should be true and should have a relatively simple proof, but I cannot figure it out.
For every $\varepsilon>0$ there ...
3
votes
1answer
171 views
A question on differential forms and integral invariants
The following question comes up in the study of metrics with the same unparameterized geodesics in Riemannian and Finsler geometry:
Question. Let $M$ be a closed manifold of dimension $2n+1$ and let ...
4
votes
1answer
157 views
Curvature for $C^{1,\alpha}$-metrics
According to Gromov's Metric Structures for Riemannian and Non-Riemannian Spaces every limit $V_0$ of sequences in the class of manifolds $V$ with $|K(V)| \leq 1$ and $\mathrm{InjRad}(V) \geq \rho ...
1
vote
2answers
139 views
Methods of probability theory in differential geometry fruitful? [closed]
I am trying to understand how the two paradigms of differential geometry and probability theory can fruitfully be applied to each other.
The more suggestive direction is to use methods of ...
18
votes
1answer
254 views
Integral formula for Euler class
The tangent bundles of closed hyperbolic surfaces have flat $PSL(2,\mathbb{R})$ connections showing that there can be no integral formula for the Euler class such connections. This contrasts the ...
1
vote
1answer
97 views
Bakry-Emery Laplacian and Hodge Decomposition
I have a question about the Hodge Decomposition theorem. Let $(X,\omega)$ be a compact Fano Kaeler manifold, we know the Hodge theorem works very well with respect to the $\bar\partial-$Laplacian ...
1
vote
0answers
160 views
Second derivative of Riemannian Exponential Map
Let $M$ be a Riemannian manifold. Let us look at the Riemannian exponential function $\exp_x: T_x M \supset \mathcal{D} \longrightarrow M$.
The derivative of the exponential map can be expressed in ...
4
votes
1answer
163 views
Boundaries of relatively hyperbolic groups
When the interior of an n-manifold $M$ has a pinched negative curvature metric of finite volume, then its fundamental group $\Gamma=\pi_1M$ is relatively hyperbolic relative to the parabolic groups ...
-1
votes
0answers
20 views
Covariant derivative along curve [migrated]
Let $M={\mathbb R}^{3} $ with the usual metric $g=ds^{2} =dx^{2} +dy^{2} +dz^{2} $. Let $\gamma :I\to M$ be a unit speed curve. How can I prove that $\nabla _{\gamma '} \gamma '=\gamma ''$ , where ...
8
votes
1answer
230 views
counterexample to the Chern number inequality on Fano manifold
We know that if an $n$-dimensional Fano manifold admits a Kahler-Einstein metric, it satisfies the following Chern number inequality
$$nc_1^n\leq 2(n+1)c_2c_1^{n-2}.$$
My question is whether there ...
2
votes
1answer
116 views
On the balanced metric on complex manifold
Let $(M,h)$ be a Hermitian manifold and denote the Kahler form of the Hermitian metric by $\omega$. From the definition of Kahler manifold we know that $M$ is a Kahker manifold if $d\omega=0$. A ...
7
votes
0answers
192 views
Surfaces with many (but not solely) closed geodesics?
Let $S$ be a closed surface embedded in $\mathbb{R}^3$,
let's say of genus zero.
I seek examples of $S$ with the following property:
If one selects a random any point $p$ on $S$, and a random
...
2
votes
1answer
87 views
Ray-Singer torsion of compact 3-manifolds with finite abelian fundamental group
Is the Ray-Singer analytic torsion for an arbitrary compact 3-manifold with finite Abelian fundamental group equivalent to the Ray-Singer analytic torsion of S^3 mod some direct product of Z_N's? It ...
3
votes
1answer
184 views
Exponential mapping versus flow
In Hamilton's article on the Nash-Moser Theorem, he gives the map that maps a vector field $X$ to its flow $e^{tX}$ in $\mathrm{Diff}(M)$ as an example where the implicit function theorem in Frechet ...
1
vote
1answer
117 views
Entropy of Negatively pinched manifolds
Suppose $M$ is a compact negatively pinched Riemannian manifold of dimension $n$. We normalize the metric such that $-1\le K\le -a^2$ for some $0<a\le 1$. Let $G$ be the fundamental group of $M$. ...
0
votes
1answer
70 views
harmonic maps from cone to $S^2$ locally lipschitz?
Are the harmonic maps from a 2-dimensional cone to $S^2$ locally lipschitz or Holder continuous?
8
votes
1answer
277 views
Are there more “types” of derivatives than “symmetric” or “alternating”?
The $kth$ derivative of a function $f:\mathbb{R}^n \to \mathbb{R}$ can be thought of as a symmetric $k$-tensor. (Well, almost. It is not invariant under coordinate changes, and should really be ...
0
votes
0answers
82 views
Does the Laplace-Beltrami/surface gradient commute with orthogonal projection? (related to Galerkin method)
Let $\Gamma$ be a $C^k$ $(n-1)$-dimensional hypersurface embedded in $\mathbb{R}^n$. Let $H=L^2(\Gamma)$ and $V=H^1(\Gamma)$.
Suppose that $\{v_j\}$ is a basis for $H$ and $V$ (not necessarily ...
1
vote
0answers
73 views
PL or projective PL map on the links of a PL manifold
Let $M$ be a PL manifold and $f: M\rightarrow M$ be a PL homeomorphism. Suppose that $f(x)=x$ for some vertex $x$. Is the restriction map of $f$ on the links of $x$ also PL? Someone claims that this ...
0
votes
0answers
135 views
Brakke's theorem for gap in entropy between self-shrinkers
In their paper The round sphere minimizes entropy among closed self shrinkers, Colding-Ilmanen-Minicozzi-White state "It follows from Brakke's theorem that $\mathbb{R}^n$ has the least entropy of any ...
