Questions tagged [dg.differential-geometry]

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

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19 views

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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21 views

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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109 views

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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103 views

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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251 views

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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115 views
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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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41 views

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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83 views

Characterization of planar domains onto which a unit disk can be mapped with constant singular values

It can be shown that there are (smoothly bounded, Jordan) domains $E\subset \mathbb{R}^2$ which are $\textit{not}$ images of mappings $f$ from the unit disk (or any other planar domain), such that $\...
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1answer
108 views

If every non null set of geodesics intersects itself in uniformly bounded finite time, is the manifold compact?

Let $M$ be a complete, connected Riemannian manifold without boundary. Given a point $p\in M$ and a subset $K$ of $S_p M$, the unit sphere in $T_p M$, define the $K$-cone of directions $C(K)$ around ...
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1answer
128 views

Curvature of principal bundle

Let $(P,M,G)$ be a principal bundle with connection 1-form $\omega$. In all books I have seen so far, the curvature is defined by \begin{equation} F:=D_{\omega}\omega \in \Omega({P,\mathfrak{g}}) \end{...
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137 views

Are there known examples of almost complex manifolds admitting neither a symplectic nor a complex structure?

I have seen the the example of $S^6$ being touted around here and there but it does not seem to be generally confirmed that there is no complex structure on it.
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169 views

How special are homogeneous spaces?

Let $M$ be a smooth finite dimensional manifold, how restrictive is it to require $M$ to admit a smooth action by a finite dimensional Lie group $G$? Related questions/approaches: Of course we need $\...
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47 views

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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29 views

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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95 views

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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151 views

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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98 views

Is there a smooth Weyl equivariant map from this quotient space into $G_2/T^2$?

It is known that $G_2$ acts transitively on $S^6$ with fibers $SU(3)$. Let us consider the following set $P$ of complex unitary $7 \times7$ matrices $A$, where $$ A = (v_0, \, v_1, \, v_2, \, v_3, \, ...
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53 views

Norm of a Taylor approximation of a multivariate function

I have a function $f:\mathbb{R}^n\to\mathbb{R}^m$. My goal is to bound the first order Taylor approximation of $f$. Given $x,x'\in\mathbb{R}^n$ I have that \begin{equation} f(x)-f(x')\approx (x-x')^...
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50 views

A characterization of functions which Riemannian Hessian equal to zero

Consider Euclidean space $\mathbb{R}^n$, and measure distances in this space with some Riemannian metric $M(x)$. That is, for two points $x, y$, define $d(x, y)$ to be equal to $$d(x, y) = \inf_{\...
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1answer
346 views

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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67 views

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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1answer
114 views

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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1answer
257 views

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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484 views
+250

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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1answer
222 views

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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162 views

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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82 views

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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39 views

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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1answer
67 views

The tangent map of the multiplication map of a vector bundle

If $\beta: \mathbf{U}\times \mathbf{V}\to \mathbf{W}$ is a bilinear map between real linear spaces then its derivative at a point $(u,v)$ is given by the Leibniz rule $$D\beta(u,v)(u_0,v_0)=\beta(u,...
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61 views

Distance function on generalized upper half planes

Let $\mathbb{H}^n$ be the quotient $GL_n(\mathbb{R})/O(n) \cdot \mathbb{R}^\times$, which at least in the theory of automorphic forms is called a generalized upper half plane. We could also think of $...
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94 views

Restrictions on pointed lifts of isometries

Let $M$ be a (closed) Riemannian manifold and let $f$ be an isometry of $M$ fixing a point $\ast \in M$ that acts trivially on $\Gamma := \pi_1(M,\ast)$. Then there is a unique isometry $\tilde{f}$ of ...
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66 views

A PDE involving a diffeomorphism of $\mathbb{S}^1$

This question is a special case of this one. Let $s(\theta)>0, b(\theta)$ be two smooth non-constant real-valued functions on $\mathbb{S}^1$. Do there exist a diffeomorphism $\phi:\mathbb{S}^1 \to \...
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127 views

Extending an embedding with trivial normal bundle

I am recently studying the book Notes on Cobordism Theory by R. E. Stong and I have noticed that the proposition below is (implicitly) used (for example to extend a $(B,f)$ structure on a boundary of ...
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1answer
735 views

Can the product of a 3-dimensional lens space with a circle be diffeomorphic to another such product when the lens spaces aren't diffeomorphic?

This is a question that I need to answer in order to resolve an issue for my dissertation and I am looking for a reference. Here is the precise statement of the question. Suppose we have two three-...
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2answers
219 views

How to find equations of a sub-Riemannian problem

I am working on sub-Riemannian geometry and try to understand what are the tools to find the equations of a sub-Riemannian problem. Here is an example: Let us consider the system defined by a ...
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96 views

Access to an old paper of Obata

I'm trying to access the following paper of Morio Obata: Conformal changes of Riemannian metrics on a Euclidean sphere, in "Differential Geometry -- In honor of K. Yano", 1972, pp. 347--353: ...
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62 views

Weights of finite abelian group actions on submanifolds/subvarieties

(cross-posted from https://math.stackexchange.com/questions/4125529/weights-of-finite-abelian-group-actions-on-submanifolds-subvarieties) How do weights associated to actions of finite subgroups of $\...
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1answer
314 views

Paths $tg_1+(1-t)g_0$ in the moduli space of Riemann surfaces

Suppose $S$ is a smooth compact oriented surface without boundary. Let $g_0$ and $g_1$ be two smooth Riemannian metrics on $S$. Consider the interpolating path of metrics $g_t=g_1t+g_0(1-t)$. Recall ...
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124 views

Does this geometric PDE have a solution?

Let $s(\theta), b(\theta)$ be two smooth non-constant real-valued functions on $\mathbb{S}^1$, and assume that $s$ never vanishes. Does there exist a map $h:(0,1) \times \mathbb{S}^1 \to \mathbb{S}^1$,...
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1answer
250 views

What is the relationship between $\mathrm{SO}(2)$ and $\mathrm{PSL}(2,\mathbb{R})$?

$\DeclareMathOperator\SO{SO}\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\R{\mathbb{R}}$The holonomy of a hyperbolic surface $S$ in terms of differential geometry is either $\SO(2)$ or $\mathrm{O}(...
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2answers
125 views

Necessary and sufficient curvature condition for a regular planar curve to be simple and closed

Given a smooth, $2\pi$-periodic function $\kappa(s)$, the associated planar curve $\gamma(s)$ for which $\kappa(s)$ is the (signed) curvature, is uniquely determined up to Euclidean invariance: a ...
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1answer
236 views

Finite self-maps exist on rigid CY3s

Let $X$ be a smooth projective rigid Calabi-Yau threefold. Question. Does there exist a finite map $X\to X$ of degree $>1$?
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32 views

On the short distance behavior of the Green functions of powers of the Laplacian

Let $M$ be a closed Riemannian manifold and let $\Delta=dd^{*}$ be the (positive) Laplacian on $M$. Given $\lambda>0$ and a positive integer $s$, set $G_{\lambda,s}=(\Delta^s+\lambda)^{-1}$. ...
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68 views

Principal bundle over associated bundle

Let $P$ be a principal $G$ bundle. Let $S$ be a space with left action of $G$, and let $Q$ be a principal $H$ bundle over $S$ with the property that the action of $G$ can be lifted to $Q$. Then $$ P \...
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135 views

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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68 views

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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0answers
132 views

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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1answer
159 views

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 ...
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1answer
70 views

Unit Killing vector fields on pseudo Riemannian manifolds

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{...
6
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2answers
267 views

Exponential convergence of Ricci flow

I've been trying to understand the asymptotic behavior of Ricci flow, and there are two facts which I am unable to square away. I'm interested in higher dimensional manifolds, but my question is ...

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