# Questions tagged [dg.differential-geometry]

Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.

8,507
questions

2
votes

0
answers

39
views

### Progess on conjectures of Palis

I came across a "A Global Perspective for Non-Conservative Dynamics" by Palis. He has some conjectures
"Global Conjecture:
There is a dense set $D$ of dynamics such that any element of ...

2
votes

1
answer

59
views

### Jacobi fields in singular metric on quotient space

Consider the square $\Omega = (0,\pi) \times (0,\pi/2) \ni (r,\theta)$ endowed with the Riemannian metric
\begin{equation}
f^2 \big(\mathrm{d} r^2 + \sin^2(r) \, \mathrm{d} \theta^2 \big),
\end{...

0
votes

0
answers

236
views

### Serre's theorem for sheaves [closed]

Serre's theorem for (algebraic) vector bundles (VB) says that a VB over an affine variety is the same thing as finitely generated projective module over the ring of algebraic functions $\mathcal{O}(V)$...

1
vote

0
answers

42
views

### Does any warped product metric with harmonic Weyl curvature admit a structure of zero radial Weyl curvature?

A Riemannian manifold $(M, g)$ has harmonic Weyl curvature iff its Schouten tensor is Codazzi, and if there exists $f: M \to \mathbb{R}$ such that $W(\bullet, \bullet, \bullet, \nabla f) = 0$, one ...

4
votes

1
answer

147
views

### Normalizer of solvable linear group is an algebraic group?

I am trying to read the article "Three-dimensional affine crystallographic groups" of Fried–Goldman (Adv. Math., 1983). At some place, it states that if $G$ is a connected solvable closed ...

1
vote

2
answers

141
views

### Exterior differential systems on compact three-manifolds and Cartan-Kähler theory

Let $M$ be a compact three-manifold. I am interested in the following equation on $M$:
$ d e^i = \sum_{j,k}^3 \Theta^i_{jk} \, e^j\wedge e^k\, , \qquad i =1,2,3$
together with the following condition:...

0
votes

1
answer

77
views

### Does any warped product metric admit a function with hessian proportional to the metric?

It is known that the existence of a function with hessian proportional to the metric implies that the metric is a warped product metric. Is the reciprocal true as well? I.e, if $(B \times N, g = g_B + ...

3
votes

2
answers

341
views

### Compactification of a product of manifolds

Let $M$ be a smooth manifold. We make the assumption that $M$ can be viewed as the interior of a compact manifold with boundary $\overline{M}$. In practice, for an explicit manifold, any ...

0
votes

0
answers

97
views

### Smooth surface with boundary equal to $N$ distinct lines [closed]

Let's start with the simple observation that if $p_1,\dots,p_N$ are distinct points on $\mathbb{S}^1$ and if $M\subset \overline{B}_1$ is a smooth embedded $1$-dimensional manifold with boundary such ...

1
vote

0
answers

61
views

### Moser iteration epsilon-regularity for non-linear system in general dimension

I am attempting to prove the following result in general dimension $n$. Given $(M^n,g)$ a Riemannian manifold with $\mathrm{Ric}_g \geq -(n-1)$ and $\mathrm{Vol}_g(B_1(x)) \geq v > 0$ for all $x \...

1
vote

0
answers

96
views

### Conformal laplacian on asymptotically flat manifolds with boundary

Let $g$ be an asymptotically flat metric on $M = \mathbb{R}^3 \setminus B_1$ where $B_1$ is the unit ball.
Suppose $X$ is a smooth vector field on $M$ that is decaying exponentially and satisfies
$$\...

1
vote

1
answer

95
views

### Lie group framing and framed bordism

What is the definition of Lie group framing, in simple terms?
Is the Lie group framing of spheres a particular type of Lie group framing? (How special is the Lie group framing of spheres differed ...

2
votes

0
answers

87
views

### The existence of a positive Green function for the Laplacian on $\mathbb R$

One can show explicitly and easily that the function $G(x,y) = \frac 1 2 |x-y|$ is a positive Green function for the Laplacian $\frac {\mathrm d ^2} {\mathrm d x ^2}$ on $\mathbb R$ (endowed with the ...

0
votes

0
answers

65
views

### A question related to frame bundle of a vector bundle [migrated]

I am currently reading up on principal fiber bundles from a set of lecture notes on the subject, and I am trying to make sense of frame bundle of a vector bundle.
Consider a vector bundle $p:E\to M$ ...

4
votes

1
answer

142
views

### Bounded covariant derivative of curvature tensor

Let $M$ be a complete Riemannian manifold.
Suppose that there are positive constants $i_0$ and $K$ such that the injectivity radius of $M$ is at least $i_0$ and $|\mathrm{Rm}|\le K$ and $|\nabla \...

1
vote

0
answers

116
views

### Minimal first Pontryagin class $p_1=1$?

From Hirzbuch theorem,
the signature of 4-manifold $\sigma = p_1/3$ with the first Pontryagin class $p_1$.
I know that the $\sigma=p_1/3 =1$ so $p_1=3$ for complex projective space $\mathbb{P}^2$.
Is ...

0
votes

0
answers

110
views

### Non-spin 3-manifold as a boundary, extension to a spin 4-manifold [closed]

What are the possible non-spin 3-manifolds $M$ whose extension of 3d $M$ to 4d $N$ can have such that $N$ be a spin 4-manifold? What are some nice examples? I want to have such a spin 4-manifold $N$ ...

1
vote

1
answer

94
views

### How do we calculate the gradient of this function defined using the Riemannian logarithm on a Riemannian manifold?

We consider the following function $\psi$ on an open subset $V\subset M,$ a Riemannian manifold of dimension $m,$ so that $\exp_p:U\to V$ is a diffeomorphism with its inverse $\log_p: V\to U$. Let $v\...

4
votes

1
answer

110
views

### Convex hull of 3 points in Cartan-Hadamard manifolds

Can the convex hull of $3$ points in a Cartan-Hadamard manifold be smooth?
A Cartan-Hadamard manifold $M$ is a complete simply connected manifold with nonpositive curvature (so it includes the ...

4
votes

2
answers

211
views

### Convergence of metric spaces of increasing dimension

Given two metric spaces we can define the Gromov-Hausdorff (GH) distance. There are compactness results stating that a sequence of manifolds of a fixed dimension, with a uniform lower Ricci bound and ...

2
votes

1
answer

106
views

### string bordism group and framed bordism group for $d \leq 6$ and $d \geq 7$

Why do the string bordism group and the framed bordism group
coincide the same in dimensions lower than 7 ($d = 0,1,2,3,4,5, 6$)?
Why do the string bordism group and the framed bordism group differ
...

3
votes

1
answer

120
views

### Derive distributional inequalities from pointwise estimates

My question is how to prove the following claim:
Suppose that $E$ is an algebraic set in $\mathbb{R}^n (n\ge3)$ with dimension $\le n-2$, and $u$ is locally Lipschitz continuous
on $\mathbb{R}^n$. If ...

0
votes

1
answer

105
views

### Decompositions of $\partial_i$ to the radial direction and rotations in higher dimensions

We know in dimension $3$,
\begin{align}
\partial_{i}= \frac{x_i}{r} \partial_{r} - \varepsilon_{ijk} \frac{x^j}{r} \frac{R^k}{r} ,
\end{align}
where $\varepsilon_{ijk}$ are Levi-Civita symbols ...

5
votes

0
answers

108
views

### Gauge Lie groupoid associated to $SO(3)$ double cover

From each Lie group $G$ and principal $G$-bundle $P \rightarrow E$ one can form an associated (or gauge) Lie groupoid as the quotient of pair groupoid by the action of $G$ on $P \times P$
$$ \frac{P \...

4
votes

1
answer

130
views

### Examples of hyperbolic manifolds of dimension $\geq$ 3 with disjoint totally geodesic hypersurfaces

I am hoping to find examples of compact hyperbolic manifolds with at least 2 disjoint totally geodesic hypersurfaces. Ideally, I would like examples in dimension at least 4, though 3-dimensional ...

3
votes

0
answers

212
views

### Critical points up to smooth homotopy

Let $M$ and $N $ be closed connected smooth manifolds of dimension $n$.
Let $f: M\rightarrow N$ be a smooth function and not null-homotopic. Is there a smooth homotopy $H: [0,1]\times M\rightarrow N$ ...

1
vote

1
answer

65
views

### Is any globally-hyperbolic manifold conformally equivalent to one with complete slices?

Let $(M,g)$ be a globally-hyperbolic Lorentzian manifold. By the seminal work of Geroch and Bernal-Sánchez, we know that
$$M=\mathbb{R}\times\Sigma,\,\,\,\quad g=-\beta^{2}dt^{2}+h_{t}$$
where $\Sigma$...

1
vote

1
answer

112
views

### Curve length in the Sasaki metric

I am trying to read Appendix II.A.2 (Distances in the tangent bundle) in Canary, Epstein, Marden (eds.), Fundamentals of Hyperbolic Manifolds: Selected Expositions and am stumbling over a calculation ...

2
votes

1
answer

80
views

### reference for reading Schoen Yau positive mass theorem proof II

I am trying to read the paper by Schoen and Yau, Proof of the Positive Mass Theorem II. The notation is very different from what I am familiar with (basically Robert Wald's book on general relativity)....

7
votes

0
answers

497
views

### What's the point of geometric representation theory?

Please forgive the provocative title, what I mean is the following:
One can find representations of Lie algebras in geometric settings, the most famous being the Bott–Borel–Weil theory. However, ...

0
votes

0
answers

85
views

### Smooth surface containing the singularity of a spatial curve

$\DeclareMathOperator\dist{dist}$Let $\gamma$ be a $C^1$ spatial curve (one dimensional real $C^1$ submanifold of $\mathbb R^3$) which except for one point $p\in\gamma$ is smooth ($\gamma\setminus\{p\}...

0
votes

0
answers

34
views

### Renormalized limit measure on a splitting Ricci limit space

I'm reading Cheeger&Colding's On the structure of spaces with Ricci curvature bounded below. I recently. In the proof of proposition 1.35 as follows. I got some questions:
Proposition 1.35. Let $\...

2
votes

0
answers

71
views

### What is known about warped product metrics satisfying conditions more general than conformal flatness?

In this paper, the authors characterize warped product metrics which are conformally flat (the fibers must have constant sectional curvature, on some cases there is a limitation on the number of ...

0
votes

0
answers

65
views

### Tangent spaces of Lipschitz sub manifolds

Consider $\mathbb{R}^n$, $k<n$, and topological embeddings (homeomorphisms onto image) $f_i : \mathbb{R}^k \supseteq B_1(0) \to \mathbb{R}^n$, $i=1,2$, which are also Lipschitz continuous and ...

0
votes

0
answers

38
views

### different definitions of holomorphic bisectional curvature

Peter Li and Jiaping Wang defined holomorphic bisectional curvature in their paper as follows:
Assume that $M^m$ is a Kahler manifold of complex dimension $m$. Let $ \{e_1, \cdots , e_m\} $ be a ...

0
votes

0
answers

40
views

### Rotational invariance of Laplace-Beltrami eigenvalue problem on smooth manifolds

I am currently looking at the eigenvalue problems of the Laplace-Beltrami operator. Let $(M,g$) be a smooth and oriented Riemann manifold. I am investigating the eigenvalue problem of the Laplace-...

3
votes

0
answers

49
views

### Covariant derivative $\nabla_{\dot{\gamma}} \dot{\gamma}$ of constrained velocity vector $\dot{\gamma}$ by distribution $B$ and bounded map $\delta_i$

I have been studying differential geometry for a while. The subject is hard to grasp at first, but gets easier once one understands the main concepts. One of them are covariant derivative $\nabla_X Y$....

0
votes

0
answers

42
views

### Sufficient conditions for chain recurrent set equal to set of non wandering points

Given a generic diffeomorphism, I know that the set of nonwandering points is contained in the chain recurrent set, but the converse is not always true. Is there some sufficient conditions under which ...

3
votes

0
answers

87
views

### Asymmetric minimal surfaces in $H^3$

Inspired by this question, are there any explicit parameterizations of asymmetric minimal surfaces in $\mathbb{H}^3$? E.g. something like the minimal surface which lies over the ellipse given by
$$y^2 ...

2
votes

0
answers

96
views

### Changing the sign of the moment map in the Seiberg Witten equations

The Seiberg-Witten equations on a closed four manifold
$$
D_A \varphi = 0, F_A^+ = \mu(\varphi)
$$
are elliptic (up to gauge transformations), and so the equations
$$
D_A \varphi = 0, F_A^+ = -\mu(\...

5
votes

0
answers

164
views

### $C^1$ manifold with complex structure

Let $M$ be a manifold. A complex structure on $M$ is an endomorphism $J \in \text{End}(TM)$ such that $J^2 = -\text{id}$ together with the vanishing of the Nijenhuis tensor. If $J$ is real-analytic, ...

0
votes

1
answer

140
views

### Going from piecewise to genuine geodesic without decreasing number of intersections?

Let $(M^2,g)$ be a complete, two-dimensional Riemannian manifold be given; also given is $\gamma: [0,\infty) \to M$, an injective geodesic in $M$.
Suppose there are two geodesic segments $\gamma_i : [...

16
votes

0
answers

363
views

### Is the oriented bordism ring generated by homogeneous spaces?

I am trying to find a Riemannian geometrically well-understood set of generators of the oriented bordism ring, including the torsion parts. By a set of generators, I mean that the set generates the ...

0
votes

0
answers

90
views

### Integration by parts over $R^n$ [migrated]

Let $\mu$ be a probability measure over $\mathbb{R}^n$. Let $f$ and $g$ be two real-valued functions on $\mathbb{R}^n$. I would like to compute
$$\int_{\mathbb{R}^n} \nabla f(x) g(x) \, d\mu(x). $$
I ...

12
votes

3
answers

920
views

### Area of a smooth complex projective curve

Let $P(X,Y,Z)$ denote a homogeneous polynomial in $\mathbb{C}[X,Y,Z]$ such that $X_P = \{(u : v : w) \in \mathbb{C}\mathbb{P}^2 \mid P(u,v,w) = 0\}$ defines a smooth complex projective curve in $\...

1
vote

1
answer

328
views

### Topological degree of differentiable map using line integrals?

Let $f:\mathbb R^2 \to \mathbb C$ be a $C^1$ function that vanishes at a point $x_0.$
I can then define $$-i \int_{\gamma_\varepsilon} \nabla \log(f(s)) \cdot ds := - i \int_0^1 \nabla (\log f)(\...

1
vote

0
answers

97
views

### Planar sections of convex sets in Cartan-Hadamard manifolds

Let $X$ be a convex set in Euclidean space $\mathbf{R}^n$ and $p\in\mathbf{R}^n$ be a fixed point. Then any plane $\Pi$ passing through $p$ intersects $X$ in a convex set. Conversely, this property ...

3
votes

1
answer

321
views

### Does Hermite-Einstein imply Kähler-Einstein?

Let $M$ be a compact Kähler manifold and let $\nabla$ be its Levi-Civita, or equivalently its Chern, connection. Denoting the vector bundle of complexified one forms of $M$ by $\Omega^1_{\mathbb{C}}$, ...

0
votes

0
answers

115
views

### Dirac distribution on a manifold $M$ as a smooth manifold in $C^1(M)^*$, question about its dimension

I have not learned many knowledges on differential geometry, I met this when trying to read the min-max scheme in PDE on manifold, which is in Section3.1.
Let $M_1= \delta_{x_i}$, $x_i \in M$. For $\|...

3
votes

1
answer

215
views

### 1D topological defects in $d>3$ spatial dimensions

I am trying to construct a 1D topological defect solution in 4 spatial dimensions, i.e., a solution to some PDE (likely the equations of motion of some Lagrangian) on $\mathbb{R}^{4}$ which is ...