Questions tagged [dg.differential-geometry]
Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.
7,217
questions
0
votes
0answers
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 ...
2
votes
0answers
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 ...
6
votes
0answers
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 ...
6
votes
0answers
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 ...
5
votes
1answer
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
...
1
vote
0answers
115 views
+100
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 ...
1
vote
0answers
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 ...
2
votes
0answers
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 $\...
2
votes
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 ...
3
votes
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{...
3
votes
0answers
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.
7
votes
1answer
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 $\...
1
vote
0answers
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 ...
1
vote
0answers
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 ...
0
votes
0answers
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 ...
7
votes
1answer
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-...
4
votes
0answers
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
$$ A = (v_0, \, v_1, \, v_2, \, v_3, \, ...
2
votes
0answers
53 views
Norm of a Taylor approximation of a multivariate function
I have a function $f:\mathbb{R}^n\to\mathbb{R}^m$. My goal is to bound the first order Taylor approximation of $f$. Given $x,x'\in\mathbb{R}^n$ I have that
\begin{equation}
f(x)-f(x')\approx (x-x')^...
0
votes
0answers
50 views
A characterization of functions which Riemannian Hessian equal to zero
Consider Euclidean space $\mathbb{R}^n$, and measure distances in this space with some Riemannian metric $M(x)$. That is, for two points $x, y$, define $d(x, y)$ to be equal to
$$d(x, y) = \inf_{\...
5
votes
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 ...
2
votes
0answers
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, ...
4
votes
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 ...
7
votes
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 ...
12
votes
1answer
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 : ...
3
votes
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 $|f''(x)|\leq M|f(x)|\quad \forall x\in \mathbb{R}$ ...
3
votes
0answers
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 ...
4
votes
0answers
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
$$\bigotimes_{i=1}^{p}T\mathcal{M}\otimes\bigotimes_{i=1}^{q}T^{\ast}\mathcal{M}.$$
...
2
votes
0answers
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 ...
1
vote
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,...
4
votes
0answers
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 $...
3
votes
0answers
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 ...
1
vote
0answers
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 \...
2
votes
0answers
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 ...
19
votes
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-...
7
votes
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 ...
2
votes
0answers
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: ...
2
votes
0answers
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 $\...
6
votes
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 ...
2
votes
0answers
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$,...
1
vote
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}(...
5
votes
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 ...
3
votes
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$?
0
votes
0answers
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}$. ...
4
votes
0answers
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 \...
3
votes
0answers
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 ...
3
votes
0answers
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 (...
2
votes
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 $\operatorname{Pol}(S^2)$ ...
2
votes
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 ...
1
vote
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
votes
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 ...
