# Questions tagged [dg.differential-geometry]

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

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### A geometric criterion for uniqueness in the Plateau problem?

Let $\gamma: S^1 \to \partial B \subset \mathbf{R}^3$ be a smooth, simple closed curve in the boundary of the unit ball. Suppose that $\gamma$ intersects every horizontal plane $\Pi_t = \{ z = t\}$ at ...
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### Consequences of Nash-Tognoli Theorem

The Nash-Tognoli theorem states that every closed and smooth manifold is diffeomorphic to a real algebraic variety. This appears to me as a very strong and surprising fact. However, I am not aware of ...
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### A strong maximum principle for varifolds of arbitrary codimension

Let $M$ be an $n$-dimensional Riemannian manifold and $N$ a hypersurface in $M$. Let $p \in N$ and $\kappa_1 \leq \cdots \leq \kappa_{n-1}$ be the principal curvatures of $N$ at $p$ with respect to a ...
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### Examples of Geometrically meaningful constraint to component of derivative similar to Cauchy Riemann equations

In most of basic Complex analysis, the geometric depth of the subject arises from the PDE that the component of derivative must obey for the function to be considered complex differentiable (/Cauchy ...
45 views

### What are known properties of the boundary curves of J-holomorphic curve with boundary

Suppose $\Sigma$ be a punctured Riemann surface with punctured boundary, and $(M, J)$ be a $2n$-manifold with almost complex structure $J$. Let $L$ be a totally real submanifolds of $M$ in smooth ...
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### Convex hulls of compact sets in a 2-manifold

Let $(\mathbb{R}^2,g)$ be a complete Riemannian manifold. Let $K\subset \mathbb{R}^2$ be a compact, connected set, and let $\text{conv}(K)$ be its convex hull, i.e., the intersection of all ...
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### Cohomology in a combinatorial way using ribbon graphs

I am interested in studying the cohomology of surfaces. Let $S$ be a compact orientable connected surface. One possible way is to learn cohomology using differential forms. Is it possible to approach ...
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### Jacobi equation and conjugate points on solution curves of the Van der Pol vector field

Let $X$ be a geodesible non vanishing vector field on a manifold $M$. Namely there is a Riemannian structure $(M,g)$ such that all integral curves of $M$ are unparametrized geodesics of the ...
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### Statements in differential geometry independent from ZFC

It is well known that some problems in functional analysis and in general topology are independent from ZFC: to name a few, Kaplansky's conjecture, the existence of outer automorphisms of the Calkin ...
66 views

### Generalizations of elliptic chain complexes

I would like to know if it is possible to generalize the notion of elliptic chain complex of differential operators to different contexts, whether geometric or non geometric. I have in mind D-geometry....
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### Intersections of geodesics in an "almost flat" plane

Let $g$ be a complete metric on $\mathbb{R}^2$, such that: Outside of a compact connected set $K\subset \mathbb{R}^2$, the curvature of $g$ vanishes. The integral of the Gaussian curvature in $K$ is ...
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### Classification of Lie group structures on $\mathbb{R}^n$

Is it possible to describe, up to isomorphism, all Lie groups $G$ whose underlying manifold is diffeomorphic to $\mathbb{R}^n$ (with its standard smooth structure)? In fact, I haven't found any such ...
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1 vote
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### Monodromy group action on de Rham cohomology

Let $f : Y \longrightarrow X := \mathbb{P}^1\setminus\{0,1,\infty\}$ be the smooth proper morphism associated to the Legendre family, which is an elliptic fibration of the punctured line, with fibre ...
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### On equipartitions of surfaces of 3D convex regions

Let S be the surface of a 3D convex region (a 'convex surface'). Let S' be a subset of S. We shall refer to S' as geodetically convex in S if the following condition holds: If A and B are two points ...
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### Six people standing on earth

Consider 6 people $p_i$, $i=1,\dots 6$, standing on a sphere $S^2$. We label the positions of these people by $p_i$ again. Suppose no pair of these points $p_i$ are antipodal. At each point $p_i$ ...
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### Is Stenzel's Ricci-flat metric on $T^*\mathbb{CP}^n$ hyperkahler?

In a well-known paper, Stenzel constructed complete Ricci-flat Kahler metrics on the total spaces of cotangent bundles of $S^n$, $\mathbb{RP}^n,$ $\mathbb{CP}^n$, $\mathbb{HP}^n$, and $\mathbb{OP}^2$. ...
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### Closed geodesics on bumpy spheres

Main question: Does every bumpy Riemannian metric on a sphere have at least three short and prime closed geodesics, for some reasonable definition of short? E.g., a geodesic $\gamma$ could be called ...
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### When is a bundle map between tangent bundles the differential of a function?

I previously asked this question on MSE but did not receive much of a response, so I'll attempt to post it here, edited with the comments received. I hope this question isn't too basic for MO ...
• 1,461
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### Further directions in representations of surface group into a Lie group

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PSL{PSL}$I studied the interpretation of Teichmuller space as a representation space for surface groups in $\PSL(2,\mathbb{R})$. Now I am planning to ...
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### Are the models of infinitesimal analysis (philosophically) circular?

Infinitesimal analysis (by which I mean that originating from topos theory---not the nonstandard analysis of Robinson) seeks to recover the pre-limit notions of calculus (which are sufficiently useful ...
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What theories are there for generalized functions (distributions) in infinite dimensions? In particular, suppose your "infinite dimensional manifold" is $\mathfrak{M}:=C^\infty(S^1)$. Is ...