Questions tagged [dg.differential-geometry]
Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.
9,275 questions
2
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Balanced + Pluriclosed implies Kähler
I was studying some complex geometry and most articles mention this result that is just a remark from an 2001 article by Alexandrov and Ivanov, called Vanishing Theorems on Hermitian Manifolds. Even ...
- 121
-1
votes
0
answers
51
views
Essential versus non-essential points in the $C^2$ isometric immersion problem: questions on exposition and proof norms [closed]
I am working on a manuscript on the local and global theory of $C^2$ isometric immersions of surfaces into $\mathbb{R}^3$. The paper is currently being prepared for journal submission, and this post ...
- 11
1
vote
0
answers
104
views
Steklov spectrum on the unit ball and its relation to the spherical Laplacian on $\mathbb{S}^3$
Let $B^4 \subset \mathbb R^4$ be the Euclidean unit ball with boundary $S^3$.
Consider the Steklov eigenvalue problem
$$
\begin{cases}
\Delta u = 0 & \text{in } \mathbb{B}^4,\\
\partial_\nu u = \...
- 569
12
votes
3
answers
1k
views
Witten confusion: supersymmetry and ... huh?
I've been working through the derivations and use of equations (13) and (15) of Witten's ‘Supersymmetry and the Morse Inequalities’. My confusion is around the operator he calls $K_j = [a^{j*}, a^j]$ ...
- 8,784
1
vote
0
answers
63
views
Taylor expansion of squared Riemannian distance under purely angular conformal change where the conformal metric does not extend
Let $(\widetilde M,\tilde g)$ be a smooth $d$-dimensional Riemannian manifold, let $x\in\widetilde M$, and set $M:=\widetilde M\setminus\{x\}$.
Let $(r,\Theta)$ denote $\tilde g$-geodesic polar ...
- 1,691
3
votes
0
answers
79
views
Regularity of the Leray Projection on the diffeomorphism group
Let $(M, \langle \cdot, \cdot \rangle)$ be a closed $C^\infty$ Riemannian manifold with $\mathrm{dim}(M) = m$ and let $\mathrm{vol}$ denote the renormalized volume measure on $M$. Let also $s > m/2$...
- 195
3
votes
1
answer
88
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Closed hypersurface of class $C^{1,1}$ must have a point of positive Gauss curvature
Suppose $S$ is a closed (i.e. compact and without boundary), connected $d$-dimensional hypersurface (that is, embedded in $\mathbb{R}^{d+1}$) of class $C^{1,1}$. Given the regularity of $S$, the shape ...
1
vote
1
answer
118
views
The normalization of a semi normal complex surface germ is a holomorphic immersion?
Given a reduced and irreducible complex surface germ $(X,0)$ which is seminormal, consider its normalization
$$
n : (\overline{X},0) \longrightarrow (X,0).
$$
Is the normalization map $n : \overline{X}...
- 135
-3
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0
answers
61
views
Conformal lightlike vector field on Lorentzian manifold
Can a conformal lightlike vector field on a closed 3-dimensional manifold be made isometric for some Riemannian metric?
- 15
6
votes
1
answer
307
views
Complete positive scalar curvature metric on $X \times \mathbb R^2$
In the survey paper "MANIFOLDS OF POSITIVE SCALAR CURVATURE" by Rosenberg--Stolz, in Proposition 7.2 they state that for any closed manifold $X^n$, the product $X^n \times \mathbb R^2$ ...
- 161
4
votes
2
answers
224
views
Are geodesically convex sets closed under intersection and diffeomorphic to $\mathbb R^n$?
Bott and Tu's book on algebraic topology proves the existence of good covers on smooth manifolds by a notion on differential topology:
Now we quote the theorem in differential geometry that every ...
- 5,565
2
votes
0
answers
40
views
Could we extend/change the notion of a marginally trapped surface in order to admit singularities?
Here a marginally trapped surface is a spacelike surface in a 4-dimensional Lorentzian manifold such that the mean curvature vector field is everywhere lightlike.
Is it absurd from a physical point of ...
- 71
4
votes
0
answers
117
views
Is there a real analytic tubular neighbourhood theorem?
The tubular neighbourhood theorem, stating that an embedded submanifold has a neighbourhood that is a diffeomorphic image of an open subset of the normal bundle, is a staple result about smooth ...
- 9,076
3
votes
1
answer
124
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Bounding norm of Jacobi fields given end point values
I posted this question on MSE first but it seems I'm not getting an answer.
I'm reading the paper on the generalized sphere theorem by Grove and Shiohama, and there is an observation they made that I'...
- 143
3
votes
0
answers
187
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Morse function for quotient of manifold?
I have a closed (compact without boundary) manifold M and a compact Lie group G that acts on it. I want to understand the topology of $M/G$, at least compute its singular homology groups. The action ...
4
votes
1
answer
352
views
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
313
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
28
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)$:
\...
- 569
1
vote
0
answers
87
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 +...
- 15
0
votes
0
answers
61
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{...
- 2,301
3
votes
1
answer
133
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 ...
- 454
1
vote
0
answers
65
views
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
62
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
125
views
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
321
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 ...
- 267
2
votes
1
answer
188
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
126
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
193
views
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 ...
- 1,691
0
votes
1
answer
79
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
153
views
Orientability of orbits
Are the orbits of an action by a simply connected Lie group on an orientable manifold necessarily orientable?
- 15
0
votes
0
answers
47
views
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
240
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
98
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
80
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
140
views
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
162
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
82
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
149
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 } \...
- 461
3
votes
1
answer
184
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?
- 7,404
3
votes
1
answer
244
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
98
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\...
- 3,442
1
vote
0
answers
138
views
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
115
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
87
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
197
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
200
views
The Hantzsche-Wendt manifold
Does the direct product of the Hantzsche–Wendt manifold and a circle admit a complex structure?
- 15
3
votes
1
answer
149
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
154
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 ...
- 195
1
vote
1
answer
311
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