# Questions tagged [dg.differential-geometry]

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

5,703 questions
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### Gauss Bonnet theorem calculation for pseudosphere

In an attempt to verify Gauss-Bonnet theorem for a Beltrami pseudosphere I calculated a simple case of the Riemann sphere. Am taking curved radius $a$ for geodesic polar co-ordinate from the smooth ...
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### Smooth Cauchy problem on a cylindrical manifold (or how to define the exponential of a differential operator)

Let $M$ be a manifold, $E \rightarrow M$ be a real or complex smooth vector bundle, and $D: \Gamma_c(M,E) \rightarrow \Gamma_c(M,E)$ be a (first order if necessary) differential operator on smooth ...
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### Degenerate Monge-Ampere equation on a bounded domain with $C^{2,1}$ boundary

In the paper by Guan Pengfei: "C^2 a priori estimates for degenerate Monge-Ampere equations" https://projecteuclid.org/euclid.dmj/1077242669 Prof. P. Guan proved in Theorem 1 that the degenerate Monge-...
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### Realizing the cross product of $\mathbb{R}^3$ as the curvature tensor of a Riemannian metric on $\mathbb{R}^3$

Is there a Riemannian metric on $\mathbb{R}^3$ for which the corresponding curvature tensor $R$ satisfies $R(X,Y)Z=(X\wedge Y)\wedge Z$? I have already discussed this question in the following post ...
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### Have complex manifolds with dual number structure on the holomorphic tangent bundle been studied?

If $M$ denotes a $2n$-dimensional real smooth manifold, then $M$ together with $J\in\operatorname{End}(TM)$ with $J^2 =\mathrm{id}$ (also called almost product structure on $M$) have been surveyed in ...
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### Differential operators on a compact Lie group associated to bracket-generating sets

Let $G$ be a compact connected Lie group of dimension $m$ with Lie algebra $\mathfrak g$. Let $\{X_1,\dots,X_h\}$ be a linearly independent set of $\mathfrak g$. Assume that $\{X_1,\dots,X_h\}$ is ...
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### Property of distance function with “smoothly” varying Riemannian metrics

Let $(M,g)$ be a smooth and compact Riemannian manifold. Suppose I have a "smoothly varying" (precisely formulating this is part of the question) one parameter family of Riemannian metrics $(M,g_t)$ ...
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### Cheeger inequality for measures

Given a probability measure $\mu$ on $\mathbb{R}^n$, its Poincare constant is the least number $C$ such that: $$\int f^2 d\mu \leq C\int \|\nabla f\|^2 d\mu$$ for all zero mean function $f$. Is ...
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### Analytic Aspects of Rational Maps

I would like some help finding references for a analytic treatment of rational maps between compact complex manifolds (that is holomorphic maps defined away from a codimension at least 2 subvariety). ...
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### A problem on real analysis related to Hilbert's fourth problem

This is an extensive re-write of a question I deleted and which had basically the same title. Identify the cylinder $S^1 \times \mathbb{R}$ with the space of (co)oriented lines in the plane by ...
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### Alternative (easier) Proof of Ambrose Singer Holonomy theorem

Let $P(M,G)$ be a principal bundle. Giving a connection on $P(M,G)$ means two equivalent things. One as an assignment of subspace of $T_pP$ for each $p\in P$ and another as a $\mathfrak{g}$ valued $1$ ...
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### Clarification needed on vector field conditions in Smale's “On gradient dynamical systems”

I previously posted the question on MSE but I haven't received an answer. I'm now posting it here in a slightly revised form. In S. Smale's, “On gradient dynamical systems,” Ann. of Math. (2), vol. ...
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### Nash isometric embedding theorem with keeping the symplectic structures of our ambient spaces

I apologize in advance if this question has an obvious answer. Let $(M,g)$ be a Riemannian manifold. Then the tangent bundle $TM$ carries a natural symplectic structure $\omega_g$. In fact $\omega_g$...
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### Periodic orbit for certain Hamiltonian on the tangent bundle

In this question a nontrivial periodic orbit is a periodic orbit which is not a singular point. Let $p: \mathbb{R}^n \to \mathbb{R}$ be a ...
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### Are $C^1$ immersions dense in $C^1$?

Let $M$ be a closed compact manifold. Is the space of all $C^1$ immersions from $M$ to $\mathbb{R}^m$ ($m> \dim M$) dense in $C^1(M; \mathbb{R}^m)$ (in the $C^1$ topology)?
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### What are hyperkähler metrics used for?

It seems that a lot of effort has been devoted to endow holomorphic-symplectic manifolds with hyperkähler metrics. It started with Calabi [4] with $T^*\mathbb{CP}^n$. Other examples include coadjoint ...
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### Theorems similar to Tischler fibering theorem

Tischler theorem states that the existence of a nowhere vanishing closed $1$-form in a compact manifold $M$ implies that the manifold fibers over $S^1$. Do you know any other diffential topology ...
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### is signed distance function real analytic for real analytic domains

If $\Omega$ is a real analytic domain in $\mathbb R^n$, is the signed distance function, $f$, defined by f(x)=\begin{cases}d(x,\partial \Omega )&{\mbox{ if }}x\in \Omega \\-d(x,\...
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### Cut on hypersurfaces and angular defects

I like very much the elementary property that if one cuts a geodesic triangle onto a sphere (one can use 3 plans that contain $0$). The cut surface of the sphere is given by the sum of the angles of ...
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### Manifolds with negative dimension – Definition, References

Does the concept of differential manifold with negative dimension make sense, in differential geometry? If yes, how is it defined? Do you have any reference to recommend? My problem was born in ...
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### Monotonicity of infimum of the Willmore energy with prescribed genus

Let $$\beta_g:=\inf\{\frac14\int_\Sigma H^2 d\mu \hspace{0.2cm} | \hspace{0.2cm} \Sigma\subset \mathbb R^{3}, \operatorname{genus}(\Sigma)=g \}$$ be the infimum of the Willmore energy of embedded ...
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### The truth about an an analogy between prime ideals and prime geodesics

I've just learnt in this Unique factorisation of prime geodesics? question that there is an analogy between prime geodesics and prime ideals in number fields. You can read more about it here: https://...