# Questions tagged [dg.differential-geometry]

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

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### Minimize the z coordinate for the intersection of two extrusions of coplanar circles along parametric paths

Sorry if my question isn't phrased as formally as it should be; this is my first math question on any online forum. I basically want to minimize the z coordinate of the intersection of some number of ...
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### Dependence of Roe algebra and coarse index on the Riemannian metric

Let $(M,g)$ be a spin Riemannian manifold. The coarse index of the Dirac operator $D$ lies in the $K$-theory of the Roe algebra, which I will denote by $C^*(M,g)$ since its construction uses $g$. I ...
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### Euclidean and Minkowski Majorana spinors - inconsistency with Wikipedia Table

In this wonderful lecture note on Clifford Algebra and Spin(N) Representations, http://hitoshi.berkeley.edu/230A/clifford.pdf Somehow I find some inconsistency with his Tables of Euclidean and ...
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### What does it mean for the torsion to blow up?

Consider the following theorem which is the main result of the Hermitian Curvature Flow paper by Jeffrey Streets and Gang Tian: Theorem 1.1. Let $(M^{2n}, g_0, J)$ be a complex manifold with Hermitian ...
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### Invariance of morse homology, doubt in proof in book “Morse Theory and Floer homology”

I am reading the book "Morse theory and Floer Homology" by Michele Audin and Mihai Damian. Now I am reading the proof of the following theorem. Link to the statement of the theorem ...
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### Question in the proof of Hilbert's theorem

I'm researching about Hilbert's theorem which says that there isn't isometric immersion of a complete surface with constant negative Gaussian curvature in $\mathbb{R}^3$. I'm taking as a reference the ...
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### What are we to deduce from a structure theorem of this type concerning totally geodesic maps?

I apologise in advance for the vague nature of the question, but some insight would be greatly appreciated. I'm reading a paper of Lei Ni concerning structure theorems for Kähler manifolds. Here is an ...
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### Fitting point on a Quadric curve

I am working on research project. I am currently using CloudCompare for my project, which calculates the Gaussian curvature and mean curvature to extract the geometric features of the points. I have a ...
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### Extending the Dirac operator on an open subset of a manifold and preserving positivity

Let $M$ be a spin manifold and $U\subseteq M$ an open ball. Let $D$ be the Dirac operator on $M$ with respect to some Riemannian metric $g$, acting on sections of the spinor bundle $S\to M$. Suppose ...
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### Existence of a certain Riemannian manifold

Notation: We denote by $\text{inj}(p)$ the injectivity radius at a point $p$ of a Riemannian manifold. By unbounded, I mean that there exist points on the manifold with arbitrarily large Riemannian ...
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### Existence of harmonic maps onto the $n$-sphere

Let $(M^n,g)$ be a closed smooth Riemannian $n$-manifold with positive scalar curvature (or positive Ricci curvature) and $(S^n, g_{st})$ be the standard round $n$-sphere. Whether there exists a non-...
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### A corollary of the non-existence of positive scalar curvature

I've been done some work with scalar curvature and managed to give a simple proof for the following result: Let $M$ be a closed manifold which do not admit a metric of positive scalar curvature. Then ...
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### Completeness on the tangent bundle

I was wondering if geodesics are defined for all time on compact Finsler manifolds, which I know very little about. In an attempt to prove this, I thought maybe I could show that if $M$ is compact, ...
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### Why the Euler characteristics of a compact connected lagrangian submanifold of $\mathbb{R}^4$ is zero?

Let's consider space $\mathbb{R}^4$ with the standard symplectic structure and let $L\subseteq \mathbb{R}^4$ be a compact connected embedded submanifold. There is a fact that if $L$ is lagrangian ...
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### Smooth morphism of smooth varieties with fibres isomorphic to an affine space

Let $X$ and $Y$ be smooth varieties over the field of complex numbers $\bf C$ (smooth integral separated schemes of finite type over $\bf C$). Let $$f\colon X\to Y$$ be a surjective morphism such ...
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### Smoothness of distance function to a compact set

Fix a non-empty compact subset $K\subseteq \mathbb{R}^n$ and let $d_K(x):=\min_{z \in K} \,\|z-x\|$ be the map sending any $x\in \mathbb{R}^n$ to its distance from $K$. Suppose that: $K$ is regular : ...
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### Differential inequalities under which a flat function must be identically zero

Let $f:\mathbb{R}\to \mathbb{R}$ be a smooth function which is flat at $0\in \mathbb{R}$. That is $f^{(k)}(0)=0,\; k=0,1,2,\ldots$. Assume that $|f''(x)|\leq M|f(x)|\quad \forall x\in \mathbb{R}$ ...
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### Union of all the entire curves in a complex manifold

Let $X$ be a connected closed complex manifold. Let $S$ be the set of non-constant holomorphic functions $f:\mathbb{C}\to X$. If $\bigcup\limits_{f\in S}f(\mathbb{C})$ is a proper subset of $X$ can it ...
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### Pseudo-tensor- and tensor-densities: Sections of what bundle?

Let $\mathcal{M}$ be a smooth manifold. A tensor field is then usually defined to be a section of the tensor bundle $$\bigotimes_{i=1}^{p}T\mathcal{M}\otimes\bigotimes_{i=1}^{q}T^{\ast}\mathcal{M}.$$ ...
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### Local coordinates of one form on a principal bundle

I am reading "Natural and Gauge Natural Formalism for Classical Field Theory" by Lorenzo Fatibene and I am really confused by his definition of a connection in local coordinates. Let's say ...
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### Research in spin geometry

I am currently learning differential geometry, but I have heard about the field of spin geometry and have skimmed through the book Dirac Operators in Riemannian Geometry by Thomas Friedrich. I have ...
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### Eigenvalues of Laplacian and eigenvalues of curvature operator

Let $(M^n,g)$ be a compact Riemannian manifold (without boundary). The symmetries of the curvature $R$ of (the Levi-Civita connection associated to) $g$ allow one to realise $R$ as a self-adjoint (...
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### Spherical harmonics, $\frak{sl}_2$, and algebra gradings

Let $S^2$ be the usual $2$-sphere considered as the quotient $S^3/S^1$, and denote by $\operatorname{Pol}(S^2)$ the algebra of polynomial functions on $S^2$. We can decompose $\operatorname{Pol}(S^2)$ ...
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### Smooth morphisms vs. submersions

This question is almost a duplicate of that question, which has a good answer. The difference is that I ask for references rather than proofs. By a reference I mean a reference to a book, or to a ...
In arXiv:math/0605371, Theorem 4 on p.8, there is the following statement: Let $X$ be a unit Killing vector field on a $n$-dimensional Riemannian manifold $M$. Then the Ricci curvature \$\operatorname{...