Questions tagged [dg.differential-geometry]
Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.
9,260 questions
4
votes
1
answer
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Difficulties in explicitly constructing the "pairing" bordism
Freed's notes give the following definition of oriented bordism.
Definition. Let $\Sigma_0$ and $\Sigma_1$ be the two oriented closed manifolds. An $ n $-dimensional bordism from $\Sigma_0$ to $\...
2
votes
1
answer
246
views
Are in- and out-boundaries of oriented bordisms determined by the orientation?
I am trying to pin down the right category of $ 2 $-dimensional bordisms that lies beneath the equivalence between $ 2 $-dimensional topological quantum field theories and commutative Frobenius ...
1
vote
0
answers
20
views
Spectral theory for the boundary operator $P_{g}^{3,b}$ arising from the Paneitz operator on the 4-ball
I am working with the following eigenvalue problem on the 4–ball $(\mathbb{B}^4,g)$, associated with the Paneitz operator $(P_g^4)$ and its boundary operator $P_g^{3,b}$ on $(\mathbb{S}^3,\hat g)$:
\...
- 551
1
vote
0
answers
80
views
Isometry between $(\mathbb{H}^2 \times \mathbb{R})$ and the Lorentz cone with Hessian metric
Is the Riemannian manifold
$(\mathbb{H}^2 \times \mathbb{R}, g)$, with
$g = g_{\mathbb{H}^2} + dt^2$,
isometric to the Lorentz cone
$$
\Lambda = \{(x, y, z) \in \mathbb{R}^3 : z > 0,\ z^2 > x^2 +...
- 21
0
votes
0
answers
52
views
Uniform upper Ricci bounds for metrics Gromov–Hausdorff close to the round sphere
Let $g_i$ be a family of smooth Riemannian metrics on the standard sphere $S^n$ ($n\geq 3$). Assume that the Gromov–Hausdorff distance satisfies $d_{GH}(g_i, g_{st}) \le 1/i$ and that
$\operatorname{...
- 2,291
2
votes
1
answer
113
views
Are there any examples of Riemannian manifolds $(M,g)$ which are conformally equivalent to $D$ but are not Gromov hyperbolic?
We know that there are Denjoy domains which are not Gromov hyperbolic but whose universal cover is conformally equivalent to the unit disk which, with the Poincare metric, is Gromov hyperbolic. So I ...
- 444
1
vote
0
answers
61
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The meaning of the initial momentum (initial covector) of a sub-Riemannian geodesic
Riemannian geodesics are determined by their initial velocity. In contrast, there may exist many different normal sub-Riemannian geodesics with the same initial velocity. But there is only one with a ...
0
votes
0
answers
52
views
Integrals over low-dimensional moving manifolds
In the context of certain stochastic interacting particle systems, I got into the following problem from differential geometry.
Setup.
Consider the two-dimensional torus $\mathbb T = \mathbb R^2/\...
- 201
3
votes
0
answers
106
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Any direct and concise way to prove that $ \Delta u+(\theta, d u)_g=f $ is the E–L equation of some functional if and only if $\theta$ is exact
I wonder if there is any direct and concise way to prove that
On Riemannian manifold, $ \Delta u+(\theta, d u)_g=f $ is the E–L
equation of some functional if and only if $\theta$ is exact ($\theta$
...
- 1,041
6
votes
2
answers
304
views
Trivialising homogeneous vector bundles over a homogeneous space
Take the sphere $S^2$ with the standard indexing of its line bundles $E_k$, for $k$ an integer. Is it true that $E_{k} \oplus E_{-k}$ is a trivial vector bundle? If so, what is the easiest way to see ...
- 257
2
votes
1
answer
168
views
Unit harmonic vector field
A vector field X on a Riemaniann manifold is harmonic if and only if 1-form metrically equivalent to X is harmonic.
Question:
On any closed 3-dimensional Riemannian manifold, is every unit harmonic ...
1
vote
0
answers
117
views
Complex Hessian comparison for Kähler manifolds with bisectional curvatures bounded from below
In Theorem 2.1 of this paper, Tam and Yu proved that:
If a Kähler manifold $M$ has ${\rm bisec}\geq2k$, $p\in M$ and $r(x)=d(p,x)$, then within the cut locus of $p$, for any unit vector $v\in T_xM$ ...
- 207
2
votes
1
answer
184
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Taylor expansion of the squared Riemannian distance where the $O(.)$ term is made explicit, and $\to 0 $ as two vectors become close
The Taylor expansion of the squared Riemannian distance is expressed by the following formula (see https://arxiv.org/pdf/1904.11860, Lemma 1 for example)
I'd like to know the specific, more refined ...
- 1,681
0
votes
1
answer
74
views
Numerically finding constrained critical points
Suppose $m, n, p, n_c\in\mathbb{N}$, $Y\in\mathbb{R}^{m\times p}$. Let $\mathcal{P}_{m, n, p} := \mathbb{R}^{m\times n}\times\mathbb{R}^{n\times p}\times\mathbb{R}^{m\times p}$ and $g:\mathcal{P}_{m, ...
1
vote
0
answers
141
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Orientability of orbits
Are the orbits of an action by a simply connected Lie group on an orientable manifold necessarily orientable?
- 21
0
votes
0
answers
45
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second order differentiability for boundary of Alexandrov space
Given a convex body $X$ in $\mathbb R^2$, by Alexandrov theorem we know that $\partial X$ is twice differentiable almost everywhere. As a result, one can define the (geodesic) curvature of $\partial X$...
- 534
1
vote
1
answer
232
views
Alternating projections between e-flat and m-flat manifolds
Consider an exponential family $E = \{ p_\theta : \forall x \in \mathcal{X},~ p_{\theta}(x) = \exp(S(x)^\top \theta - F(\theta)), A \theta = b \}$ affinely constrained in its natural parameter and let ...
- 247
4
votes
0
answers
95
views
Is area controlled by total absolute Gauss curvature?
Consider a smooth closed hypersurface contained in the unit ball in $n$-dimensional space. Can its $({n-1})$-volume be bounded by a function of the integral of its absolute Gauss curvature?
- 2,966
2
votes
0
answers
77
views
Is there a Laplacian comparison theorem for Busemann functions?
Let $\gamma \colon [0,+\infty) \to M$ be a ray on a complete non-compact manifold $M$. We define the Busemann function (w.r.t. $\gamma$)
$$ \beta_\gamma = \lim_{t \to \infty} (d(x,\gamma(t)) - t). $$
...
- 21
2
votes
1
answer
136
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Integrability of direct sum of distinct Ricci eigendistributions
Some work I have been doing has hit a wall which apparently can only be breached if (among other things, this being one of them) I prove that the direct sum of any two distinct Ricci ...
- 811
1
vote
0
answers
155
views
Real structure of dual vector bundle
Suppose $V$ is a holomorphic vector bundle on a Riemann surface X which is endowed with an anti-holomorphic involution $\sigma_X: X\longrightarrow X$. $V$ is said to have real structure if $\exists$ ...
- 39
3
votes
0
answers
81
views
Product decomposition chart simultaneously adapted to $k>2$ complementary integrable distributions
Let $M^n$ be a smooth manifold and suppose $D_1, \ldots, D_k$ are integrable complementary distributions, i.e, $TM = \displaystyle \bigoplus_{1 \leq i \leq k} D_i$. Let's call the corresponding ...
- 811
0
votes
1
answer
147
views
Harmonic map flow on the sphere going to the north pole as $|x| \to \infty$
My question concerns the harmonic map heat flow from $\mathbb R^2$ onto $S^2 \subset \mathbb R^3$, given by
\begin{equation}
\begin{cases}
\partial_t u = \Delta u + |\nabla u|^2 u & \text{in } \...
- 451
3
votes
1
answer
180
views
Existence of harmonic coordinates on Lipschitz manifolds
Suppose that we have a Riemannian manifold $M$ whose metric is only Lipschitz. Does $M$ admit harmonic coordinates?
- 7,394
3
votes
1
answer
228
views
Geometric insights from the computation of algebraic-topological invariants
(This is a sibling question of the "inverse" implication)
First, I see differential topology as a bridge between more "geometrical" data/structures/methods and more "...
- 560
0
votes
1
answer
95
views
Is every submanifold the zero of a sufficiently regular section of a vector bundle on some neighborhood of the submanifold?
Let $M$ be an $m$-manifold and $N\subseteq M$ an (embedded) $n$-submanifold $(0<n<m)$. Everything in this question is assumed smooth if relevant and not stated explicitly otherwise.
Let $\pi:E\...
- 3,432
1
vote
0
answers
136
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Existence of positive solutions of linear PDE on closed manifolds
If I have a linear PDE on closed manifold
$$
L u=-\Delta u+\langle d u, X\rangle+h u=0,
$$
where $ h \in C^{\infty}(M) $ and $X$ is a 1-form which is not exact.
I wonder if there is any sufficient ...
- 1,041
4
votes
0
answers
111
views
Curvature characterization of Kodaira–Iitaka dimension
Let $X$ be an $n$-dimensional complex manifold and let $L$ be a holomorphic line bundle over $X$. We denote by $\kappa(L):=\kappa(X,L)$ the Kodaira–Iitaka dimension of $L$.
In the extreme case, say, $...
- 335
1
vote
0
answers
49
views
Connection induced by pull-back of conjugate bundle
Suppose X be a Riemann surface endowed with anti-holomorphic involution $\sigma_X$. $V$ is a holomorphic vector bundle on $X$ with holomorphic connection $D$. It is a fact that $\sigma_X^*\overline{V}$...
- 39
2
votes
0
answers
84
views
On the “canonical” torsion‐free bi-invariant connection ∇ₓY=½[X,Y] and its relation to commutator flow ½-term
Let $G$ be a Lie group with Lie algebra $\mathfrak g$. Consider the connection $\nabla$ on $TG$ defined on left‐invariant vector fields $X,Y$ by
$
\nabla_XY \;=\;\tfrac12\,[X,Y].
$
It is well known ...
- 95
2
votes
0
answers
196
views
Hyperbolic affine 3 dimensional manifold
Assume that M is a compact 3-dimensional affine manifold which is also a Seifert manifold with vanishing Euler number and whose base orbifold has negative Euler characteristic. How to prove that the ...
1
vote
0
answers
188
views
The Hantzsche-Wendt manifold
Does the direct product of the Hantzsche–Wendt manifold and a circle admit a complex structure?
- 21
3
votes
1
answer
146
views
Low-rank extensions of the inclusion function
Let $\mathcal M$ be a smooth compact $m$-dimensional submanifold (with or without boundary) of $\mathbb R^n$. I am interested in functions $f: \mathbb R^n \to \mathbb R^n$ which extend the inclusion $\...
- 230
3
votes
1
answer
147
views
Smoothness of the differential on the group of diffeomorphisms over a compact Riemannian manifold
Let $(M, \langle \cdot, \cdot \rangle)$ be a closed Riemannian manifold (compact without boundary) and consider $\mathcal{D}^s$ the group of diffeomorphisms from $M$ to $M$ of class $H^s$ ($s$-th ...
- 165
1
vote
1
answer
304
views
Can we write the Einstein tensor as exterior derivatives of differential forms?
In general relativity textbooks a conserved quantity is a tensor $J^\mu$ that satisfies $\nabla_\mu J^\mu=0$ (with $\nabla$ the Levi Civita connexion associated to a metric $g$). One can also write ...
- 4,517
5
votes
1
answer
120
views
The definition of Q-curvature on $(S^3,g_0)$ and general definition $n\geq 1$
I am reading the paper "On Conformal Deformations of Metrics on $\mathbb{S}^n$" by Juncheng Wei and Xingwang Xu, and I am trying to understand how equation \eqref{1} is derived. The authors ...
- 551
5
votes
1
answer
386
views
How can one define the Lie bracket of two foliations?
In this question I am inspired by a recently closed MO question who tried to define a kind of Lie bracket on the space of 1 dimensional singular foliations.
However that idea had a gap but I think the ...
- 330
3
votes
0
answers
95
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Adjoint space of a compact irreducible symmetric space
From Helgason's book [1], Ch. VII, $\S10$, page 326-327:
Let $(\mathfrak u,\theta)$ be an orthogonal symmetric Lie algebra of the compact type and suppose $\mathfrak k_0$, the fixed point set of $\...
- 2,209
3
votes
0
answers
83
views
How to generalize the notion of distance between sections to open manifolds as done for h-principles?
In Introduction to the h-principle by Cieliebak, Eliashberg and Mishachev (so the second edition), the authors state on page 26 the holonomic approximation theorem. This states that (I'm shortening ...
- 31
-1
votes
1
answer
175
views
When can we define "distance" for pseudo-riemannian manifolds?
You cannot define a distance functional on a pseudo riemannian manifold due to the existence of "negative directions". In Lorentzian geometry, we can fix this by computing distance along ...
- 11
1
vote
0
answers
124
views
Direct limit of infinite-dimensional manifolds
In H. Glöckner's paper Direct limit Lie groups and manifolds (J.Math. Kyoto Univ. 43-1 (2003), 1-26), it is proven that, given an ascending sequence of finite-dimensional smooth manifolds $M_1 \...
2
votes
1
answer
202
views
Area growth of disks
I consider $g$ a Riemannian metric on the unit disk $\mathbb{D}_1(0)\subset \mathbb{R}^2$. Under which geometric assumption on $g$, is there a quantitative estimate on a small number $\delta>0$ and ...
- 625
5
votes
1
answer
208
views
Necessary and sufficient conditions for a closed differential form to be the pullback of a volume form?
Considering $\alpha$ a $k$-form on a manifold $X$, $\text{dim}\,X=n> k$, when can we say that $\alpha$ is the pullback of a volume form ? (ie there exists a map $\phi : X\rightarrow Y$ with $Y$ ...
- 4,517
2
votes
0
answers
94
views
Bounds on the angular derivatives of the induced metric in polar coordinates
Let $(M, g)$ be a compact Riemannian manifold of dimension $n$ whose (non-constant) sectional curvature $K$ is such that
$$
-a^2 \leq K \leq b
$$
for some constants. Let $p \in M$ be fixed and let us ...
- 85
3
votes
0
answers
77
views
Futaki invariant as weight
The definition I studied of Donaldson Futaki invariant is the following. If $X$ is a Fano manifold with Kahler form $\omega$ , any holomorphic field can be expressed in the form $Z=(\bar\partial f)^\...
- 131
5
votes
1
answer
352
views
Functors from category of smooth manifolds to category of symplectic manifolds
Let $Man_n$ be the category of connected smooth n-dimensional manifolds with morphism being local diffeomorphisms. Let $Symp_{2n}$ be the category of connected symplectic 2n-dimensional manifolds with ...
- 111
0
votes
1
answer
92
views
Amsler surface to a pseudospherical surface of revolution
Consider an Amsler surface—that is, a pseudospherical surface containing two straight lines that intersect at a point on the surface. It is known that such a surface is uniquely determined by the two ...
- 247
1
vote
0
answers
64
views
First Dirichlet eigenvalue $\lambda_1(B(z,R))$ in the hyperbolic space
Let us consider the hyperbolic space $\mathbb{H}^n$. Given $z \in \mathbb{H}^n$ and $R > 0$ let us consider the bottom eigenvalue $\lambda_1 = \lambda_1(B(z,R))$ of the Neumann problem
$$
\Delta u +...
- 73
2
votes
1
answer
349
views
Question about Mok's paper
I have a question about Proposition 3.3 of Mok's 1990 Math Ann paper An embedding theorem of complete Kahler manifolds
of positive Ricci curvature onto quasi-projective varieties. When Mok says $F\...
- 3,433
4
votes
0
answers
107
views
Green function of the Dirac operator on spinor bundle
I am working with the following definition for the Green function of the Dirac operator $D$ on a spinor bundle $\mathcal{S}$ over a closed Riemannian manifold $(M^{n},g)$.
Let $\pi_{1},\pi_{2}:M \...
- 41