# Questions tagged [dg.differential-geometry]

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

5,708 questions
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### Geometrical regularity of the projection/normalization of a curve

Let $v:\mathbb{R} \rightarrow \mathbb{R}^3/{(0,0,0)}$ be a $C^\infty$ regular arc-length parametrization of a space curve. W.l.o.g. let us assume $v(0)=(1,0,0)$, $v'(0)=(1,0,0)$. Let $\bar{v}$ be ...
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### Interpretation of the Schouten bracket as an integrability condition

The Schouten bracket is an extension of the Lie bracket to multivector fields. Given a multivector field $\Lambda$ the vanishing of the Schouten bracket $[\Lambda,\Lambda]=0$ is referred to as a sort ...
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### Covering S2 with great circles

Let $S_2$ be the unit 2-dimensional sphere. Is there a way to cover it with great circles such that each point on $S_2$ has 1 or 2 great circles that go through it?
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### Interior gradient estimate of mean curvature equation

I'm studying by myself Mean Curvature Flow and I'm reading the paper "Interior estimates for hypersurfaces moving by mean curvature" by Klaus Ecker and Gerhard Huisken. I'm stuck in the theorem $2.3$. ...
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### Simple application of Bochner--Reilly--Weitzenböck type formulas

I am looking for simple (but not worn-out) application of Bochner--Weitzenböck type formulas in comparison geometry. (I want to use it as a motivation for students.) The vanishing theorems and ...
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### Existence of a certain kind of compact spin manifold with boundary

For a compact spin Riemannian manifold $(M^n,g)$ without boundary, $n \not\equiv 3\mod 4$, it is well-known that the Dirac operator associated with a fixed spin structure $S\rightarrow M$ has real, ...
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### What is the geometric meaning of one Riemannian metric bigger than the other one on a smooth manifold?

Gromov conjectured in 1985 and LLarull proved in 1998 that: If $g > g_0$ on the sphere, then there exists some point p on the sphere with $Sc(p) < Sc_0(p)$. Here $g, g_0$ are Riemannian metrics ...
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### The de Rham complex of the octonionic projective spaces

The complex projective space $\mathbb{CP}^n$ is a complex manifold, and hence its de Rham complex carries a representation of the complex numbers in the form of its complex structure. The quaternionic ...
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### Relation between Optimal Transport Cost and Difference between Topological Invariants?

I was working on some mathematics of Wasserstein GAN and found out a seemingly interesting research problem but I am not quite sure whether it has already been studied in some recent literature of ...
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### Diagonal is representable then composition is representable

Let $\mathcal{X}$ be a stack over $S$ i.e., a stack over category of schemes over $S$ (which we denote by $Sch/S$) which comes with a functor $\mathcal{X}\rightarrow Sch/S$. Consider the diagonal map ...
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### Vector-valued forms in Riemannian geometry

Suppose $(M,g)$ is a Riemannian manifold. I want to find a vector-valued $2-$form $T$ such that, for any vector fields $X,Y,Z$ on $M$, $$g(T(X,Y),Z)=g(T(Z,X),Y)\,.$$ As a motivation, consider the ...
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### Diagonal is representable then any morphism is representable

Ariyan Javanpeykar said here in comments that, If the diagonal is representable, then isn't any morphism $S\rightarrow \mathcal{X}$ with $S$ a scheme representable? I could not find the statement (...
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### Existence of solution for the PDE $p \frac{\partial f(p, q)}{\partial p}-q \frac{\partial f(p, q)}{\partial q}=g(p, q)$

Consider the following PDE: $$p \frac{\partial f(p, q)}{\partial p}-q \frac{\partial f(p, q)}{\partial q}=g(p, q),\tag{\star}$$ where $g$ is a flat function at the point (...
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### The developing map of conformally flat manifold

There is one sentence I don't understand in some paper. "A simply connected and conformally flat three mainifold can be conformally immersed into $S^3$" by the means of a developing map. Is any ...
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### Fourier transforms and nontrivial vector bundles

We know that in arithmetic, geometry and analysis, Fourier transforms of various forms show up. For example, we have the classical Fourier transform, Fourier-Mukai transforms in the setting of ...
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### A Generalized Bernstein's Problem

Suppose $\Sigma\subset \mathbb{R}^n$ be an area-minimizing hypersurfaces, the standard results in GMT tell that when $3\leq n\leq 7$, $\Sigma$ is a hyperplane; while for $n\geq 8$, there are examples ...
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### Linearization of hamiltonian torus action

Let $(M,\omega)$ be a symplectic manifold with a hamiltonian effective torus action. Suppose it has an isolated fixed point $p$. Is it true that there exists an invariant neighborhood $U$ of $p$ such ...
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### Sectional curvature of leaves of foliation

Given a $k$- dimensional foliation $F$ of a riemannian $n$-manifold $M$, with the property that the leaves of the foliation have constant sectional curvature $s$, for some $s$, is it true that $M$ ...
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### Deforming metrics from non-negative to positive Ricci curvature

Given a closed Riemannian manifold $(M,g)$ with non-negative Ricci curvature and $dim\geq 3$, when can we deform the metric to a positive Ricci curved one? I know it's impossible in general due to ...
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### Equivalence of two definitions of jets of smooth functions

In the literature I have encountered two different definitions of jets of smooth functions, and I was wondering how one could identify these definitions. One definition is the often encountered ...
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The following theorem is commonly attributed to Jacques Hadamard. Assume $\Sigma$ is a smooth locally convex immersed surface in the Euclidean space. Then $\Sigma$ is embedded and bounds a convex ...
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### Smooth functions on sphere

Let $u$ be a smooth function defined on the unit sphere $S^2$. Assume $u$ has two local maxima, two local minima, and two saddle points (a total of 6 critical points). Does there exist a plane $P$ ...
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### Is this pullback of non-degenerate form invertible?

For a fiber bundle $M\longrightarrow N$ where $\dim N=n$, a non-degenerate 1-form $\theta$ on $M$ generates the differential ideal $\mathcal{I}$, and the Lagrangian $\mathcal{L}$ is an $n$-form on $M$....
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### Eta-Invariant and Atiyah-Patodi-Singer Index Theorem

In Quantum Field Theory and Jones Polynomial (equation 2.16), Witten used a formula relating the APS eta-invariant to the Chern-Simons action. Witten claimed that it is derived from the Atiyah-Patodi-...
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### Maximally symmetric affine manifold

As a physicist who knows (something) about General Relativity, I'm accustomed to the term "maximally symmetric space" being an $n$-dimensional manifold with $\frac{n(n+1)}{2}$ Killing vectors. A ...
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### Research trends on mean curvature flow

It's been a few months since I've been curious about mean curvature flow and now I'm reading Robert Haslhofer's lecture notes. I like the subject and would like to do research on it, but I know ...
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### Killing vector fields of a conformally flat Riemannian metric

Let $f: \mathbb{R}^n \to \mathbb{R}$ be a smooth function and let's consider the conformally flat Riemannian metric $g = e^f \delta_{ij} dx^idx^j$ on $\mathbb{R}^n$. Is it true that the Killing ...
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### Local planar sections of surface are circles then surface is a sphere?

Is the following true: Let, $S$ is a compact connected $(n-1)$-dimensional surface in $\mathbb{R}^n$ s.t., for every point $p \in S$ there is a neighborhood $\mathcal{V}_p \subset \mathbb{R}^n$ of ...
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### Trace free Codazzi Tensor on Hyperbolic manifolds

Does there exist trace free nontrivial symmetric Codazzi Tensor on closed manifold with constant sectional curvature -1? I know, locally all Codazzi Tensors on closed manifold $(M, g)$ with constant ...
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### An upper bound for the number of singularities of a transversal vector field isometric to the zero field

Let $(M,g)$ be a Riemannian manifold. We equip the tangent bundle $TM$ with the Sasaki metric $g_s$. A smooth vector field $X:M \to TM$ is called a transversal vector field if $X(M)$ is transverse ...
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### Grassmannians of planes isotropic with respect to general tensors

In symplectic geometry, the Grassmannian of isotropic planes for a symplectic vector space is a well known and well studied object; for example, one can realize it as a homogeneous space with a known ...
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### Explaining patterns in modular multiplication graphs

Let the multiplication graph $n/m$ be the graph with $m$ points distributed evenly on a circle and a line between two points $a$, $b$ when $an \equiv b\operatorname{mod} m$. These graphs often look ...
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### Embedding of $CP^2/CP^1$ into euclidean space [closed]

Is there a "nice" embedding of $\mathbb{C}\mathbb{P}^2/\,\mathbb{C}\mathbb{P}^1$ into $\mathbb{R}^8$?
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### Positive Ricci curvature on fiber bundles

My advisor and I are working on Ricci curvature and an anonymous referee pointed out the following conjecture: Let $F\hookrightarrow M\stackrel{\pi}{\to}B$ be a fiber bundle from a compact manifold ...
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### Geometry of Affine Kac-Moody Algebras

I recently asked this question on phys.SE and it was suggested to me to ask it here. One can reconstruct the unitary irreducible representations of compact Lie groups very beautifully in geometric ...
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### The converse to the positive mass theorem

Let $(M^n,g)$ be an asymptotically flat manifold of decaying-order $\tau>\frac{n-2}{2}$, the positive mass theorem states that if the scalar curvature $S_g$ is non-negative, then the ADM mass $m_g$...
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### Einstein warped product manifold Ricci flat

Let $(M,g)=(N,\ddot{g})\times f(B,\bar{g})$ be an Einstein warped-product manifold Ricci flat (i.e. $Ric=\lambda g$ with $\lambda=0$) where $f:N \rightarrow (0, \infty)$ (positive scalar function) and ...
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### The Deligne-Mumford Compactification for Closed Surfaces

I am reading this note on super-Riemann surfaces. In the second paragraph of section 7.4.1 (page 87), there is a statement that I am trying to understand: The compactified moduli space of closed ...
A Weyl connection on a smooth $n$-manifold $M$ is a torsion-free connection $\nabla$ on its tangent bundle that preserves some conformal structure $[g]$ on $M$. By this I mean that its parallel ...