# Tagged Questions

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

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Let $M$ be an n-dimensional manifold endowed with a Riemannian metric. Suppose we have a coordinate chart, say $(U,\varphi)$ where $U\subset M$ and $\varphi\colon U\rightarrow {\mathbb R}^n$, and let $... 1answer 91 views ### On the$\omega$-limit set of a trajectory converging to a submanifold Let$X$be a$C^1$vector field on$\mathbb{R}^n$. Let$S$be a compact submanifold of dimension$s(<n)$. Suppose$S$is invariant under the flow of$X$and that we know everything about the ... 0answers 111 views ### Aityah-Patodi-Singer theorem in odd dimensions and Maslov triple indices Let$W$be a compact manifold with boundary and$D^W$a graded Dirac type operator on$W$, of product type near the boundary acting on a vector bundle$E\to W$. One obtains a graded Fredholm operator$...
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I have a question in the field of currents: Let M be an n-dimensional smooth manifold, and let T be a k-current (induced by a k dimensional sub-manifold), I would like to approximate it by a series of ...
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### Techniques to solve a non linear differential equation related to curvature

Many years ago, I considered the following non linear differential equation: $y=y''.(1+y'^{2})^{-3/2}$ This equation expresses the equality between the value of a given function $y\in C^{2}(R)$ and ...
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### Semistability of a sheaf on nodal curve

Suppose $X$ is a projective, connected, nodal curve (can be reducible) over an algebraically closed field $k$ of arbitrary characteristic. Let $F$ be a pure sheaf on $X$ and denote by $\pi^{*}(F)$ its ...
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### List of Applications of the $\partial\overline{\partial}$-lemma

Quoting from Huybrecht's book Complex Geometry on the $\partial\overline{\partial}$-lemma for Kaehler manifolds: Although it looks like a rather innocent technical statement, it is crucial for ...
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### Does a Kähler manifold always admit a complete Kähler metric?

Every smooth manifold admits a complete Riemannian metric. In fact, every Riemannian metric is conformal to a complete Riemannian metric, see this note. What about in the Kähler case? Does a ...
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### Question on Harmonic maps between Riemannian manifolds

In Theory of harmonic maps, main goal is to find minimum of Dirichlet energy function which is defined as follows: $$E(f):=\frac{1}{2}\int_M\|df\|^2dvol_g\qquad f:(M,g)\to(N,h).$$ In many Books such ...
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### On two functions with isodirectional gradients

Let $U\subset \mathbb{R}^n$ be open and $f,g:U \to \mathbb{R}$ be two $C^1$ functions whose gradients are always in the same direction, i.e. $\forall i,j \in \left\{1,...,n\right\}$ (\...
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### Two questions about Li-Yau-Hamilton estimate

This question is from my question on mathematics. Picture below is from 231 page . For to prove $Q\ge 0$ on $M \times (0,T)$, $(\partial_t -\Delta)Q \ge 0$ and $Q\ge 0$ are needed to prove.But I ...
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### How to tell if it's a Moishezon morphism

Suppose that $f \colon X\rightarrow S$ is a proper morphism of reduced and irreducible complex spaces and $f$ is a smooth deformation in the sense of Kodaira and Spencer. If we know each fiber $X_s$, ...
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### Covering rough boundaries of closed sets in manifolds by charts

This question is a little vague, I'm afraid, because I'm not sure I expect there to be a complete answer; but there should be some sort of situations where it is possible. Consider a Riemannian ...
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### Can infinitely many orbifolds be “added up” to form a fractal space?

Disclaimer: this question is rather vague and thus might not be suitable for this site. If so, feel free to tell me and I'll delete it. Intuitively, an orbifold, from what I understand, is a "...
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### Uniform approximation of a continuous flow by a $\mathcal{C}^1$ flow

Setup: Consider a (smooth) compact Riemannian manifold $M$, whose distance is denoted by $d$. Let $\Phi$ be a continuous flow, namely a continuous application from $\mathbb{R} \times M$ to $M$ ...
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### Interior periodic points of area preserving homeomorphisms of a pair of pants

A celebrated result of Franks shows that any area preserving homeomorphism of the closed annulus $A$ with at least one periodic point (possibly along the boundary) has infinitely many interior ...
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### On a parallelizable manifold, is there always a frame satisfying $[X_i,X_j]=0$?

[This question was asked on MSE, but got no answers, I thought it could be more appropriate here] Let $M$ be a parallelizable manifold. Is there always a global frame $(X_i)$ such that $[X_i,X_j]=0$...
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### Applications of Hessian operator in the Riemann manifold. Simple samples $S_{2}(f)$

Study article R. C. Reilly is entitled Applications of Hessian operator in the Riemann manifold had a doubt in the remark, shortly after the theorem 2 of that Article. The theorem is stated as: ...
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### Open non-parallelizable 4-manifolds

Let $M$ be a connected orientable open 4-manifold (noncompact, without boundary). Is it possible for $M$ to be non-parallelizable ? If yes, what example of such $M$ is there ? [EDIT : The answer ...
In this Paper there is a proof that a closed plane curve of length $L$ and curvature bounded by $K$ can be contained inside a circle of radius $L/4 - (\pi - 2)/2K$. Are there similar results for ...