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

5
votes
1answer
42 views

Condition on a differential form arising from the theory of elasticity

Let $D$ be the unit $n$-ball (for concreteness). Let $\beta\in\Omega^1(D;R^n)$ be an $R^n$-valued one-form, having full rank (viewed as a section of $T^*D\otimes R^n$). Under what conditions on $\beta$...
2
votes
2answers
68 views

Can we specify the value of harmonic forms at a point?

Let $M$ be a smooth $d$-dimensional oriented Riemannian manifold, and let $1 < k < d$ be fixed. Let $p \in M$, and let $\alpha_p \in \bigwedge^k(T_pM)^*$. Does there exist an open ...
3
votes
0answers
156 views

Differential geometry and category theory

Does anyone know of a book/paper/anything, the longer the better introducing differential geometry from a category theoretic point of view? Everywhere it seems categorical language is the elephant in ...
4
votes
0answers
40 views

Higher order variations of Riemannian geodesics

Consider a mapping $\Gamma$ from the Euclidean plane or an open subset to a Riemannian manifold $M$, so that each $\Gamma(s,\cdot)$ is a geodesic. There is a well established theory of the first order ...
-4
votes
0answers
30 views

Focal point (Definition ) [on hold]

I Am a bigginer in the differential geometry And I need the definition of focal points, all the books i see is defined in riemannian submanifolds with jacobi fields ... I know just the notions of ...
4
votes
0answers
73 views

$\ell_p$ geodesic distance on smooth Riemannian manifold and Logarithmic Sobolev Inequalities

Bear with me, I'm not a professional geometer. Recently, I've been studying Logarithmic Sobolev Inequalities (LSI) for probability distributions on manifolds (e.g as done in works of Bobkvo et al. ...
0
votes
1answer
80 views

Is $TS^2\setminus Z$ a $S^2$- fibre bundle on the puntured plane?(Swapping the role of fibre points and base space)

Let $X=TS^2\setminus Z$ where $Z$ is the zero section of the tangent bundle of $S^2$. Is there a $S^2$- fiber bundle structure on $(X,\mathbb{R}^2\setminus\{0\},q)$ for some continuous fibre map $q$?
0
votes
0answers
323 views

Finding a metric in $ \Bbb R^2 $ depending on $s$ such that $x^s+y^s=1$ is a geodesic for all $s$ wrt. the metric [migrated]

Looking for a Riemannian metric or Pseudo-Riemannian metric in $ \Bbb R^2 $ depending on $s$ such that $x^s + y^s = 1$ is a geodesic for all $s$ wrt. the metric. $x,y\in(0,1), s\in \Bbb R(0, \infty). $...
1
vote
0answers
48 views

Is there an analog of a Chern-Simons formula for the pfaffian $Pf(F)$ of a $SO(2n)$ curvature $F$?

..something similar to $tr(A \wedge dA + 2/3 * A \wedge A \wedge A)$ for $n = 2$ ?
7
votes
0answers
254 views
+50

The differential of the Gauss normal map from a Lie algebraic view point

Let $S\subset \mathbb{R}^3$ be a smooth surface with the Gauss normal map $N:S\to S^2$. Then for every $x\in S$, the differential $(dN)_x:T_xS\to T_{N(x)}S^2$ can be considered as an ...
1
vote
0answers
58 views

Does this chain rule in Sobolev spaces hold?

Let $\Omega \subseteq \mathbb{R}^n$ be an open bounded set. Let $S \subseteq \mathbb{R}^k$ be an open dense smooth submanifold of $\mathbb{R}^k$. Let $u \in W^{1,p}(\Omega,\mathbb{R}^k) \cap C(\...
3
votes
2answers
87 views

area variation of a closed surface under ${\rm SL}(3)$

Let $\Sigma$ a closed oriented embedded surface in $R^3$. When $\Sigma$ is a round sphere, then for any smooth curve $A(t) \in SL(3)$ through the identity (i.e. $A(t) \in R^{3\times 3}$, $\det(A(t))=1$...
8
votes
5answers
308 views

Reference request: Recovering a Riemannian metric from the distance function

Let $M = (M, g)$ be a Riemannian manifold, and let $p \in M$. Writing $d$ for the geodesic distance in $M$, there is a function $$ d(-, p)^2 : M \to \mathbb{R}. $$ This function is smooth near $p$. ...
4
votes
0answers
119 views

Deforming a section to a section without zeros

Let $M$ be an oriented manifold of dimension $n$. Suppose furthermore that $E$ is an oriented vector bundle of rank $n-1$ over $M$. Let $s$ be a section of $E$ transversal to the zero section in $E$. ...
9
votes
1answer
203 views

Fourth obstruction, Pontryagin and Euler class

Assume the first three obstruction classes of a rank 4 vector bundle vanish and look at the fourth obstruction class. This fourth obstruction class can be decomposed as the Euler class and the first ...
3
votes
1answer
68 views

Codimension reduction for developable Euclidean submanifold

Let $U^{m} \subset \mathbb{R}^{m}$ be an open set. Suppose $\varphi$ is an immersion of $U^{m}$ into $\mathbb{R}^{m+n}$ satisfying the following condition: For each point $p \in \varphi(U^{m})$, the ...
6
votes
0answers
94 views

One question about the $\eta$ invariant

This question is from the paper, The Analysis of Elliptic Families II. Dirac Operators, Eta Invariants, and the Holonomy Theorem, Commun. Math. Phys. 107, 103-163 (1986) --- Proposition 2.8. Suppose ...
3
votes
0answers
98 views

Diffeomorphism group action on the space of embeddings

Let $S$ and $M$ be two finite-dimensional smooth manifolds with $\dim S\le \dim M$. Then it is known (e.g.Kriegl-Michor's book) that the set $\mathrm{Emb}(S, M)$ of all smooth embeddings $S\to M$ is ...
5
votes
1answer
88 views

Can I bring the Kirillov 2-form on coadjoint orbits to adjoint orbits?

I tried asking this question on stackexchange and received no response. Given a semisimple Lie group, there is a symplectic structure on the coadjoint orbits arising from the Kirillov 2-form. Can I ...
1
vote
0answers
55 views

Hodge inner product and orientability [duplicate]

I am writing some thing about Hodge theory. On the references I am reading, is seems that when we define the Hodge star and Hodge inner product, the orientability of the manifold is not mentioned at ...
7
votes
1answer
183 views

Atiyah-Patodi-Singer for manifolds with cusps

Dear Colleagues and Friends, Please let me know if you are aware of any references to the following question. The classical result of Atiyah, Patodi and Singer tells us that if $W$ is a compact ...
3
votes
2answers
422 views

Curl as a divergence… Is it possible? [closed]

I want to know if it is possible to express the operation $$ \nabla \phi \times (\nabla \times \mathbf A) $$ as the divergence of second order tensor field $T$. Here $ \phi$ is a scalar field and $\...
3
votes
0answers
109 views

Cohomology of the Hilbert square of a degree 14 K3 surface [Beauville-Donagi]

I have a question about the article by Beauville-Donagi called La variété des droites d'une hypersurface cubique de dimension 4 (C. R. Acad. Sc. Paris, t. 301, Série I, n° 14, 1985). Their ...
0
votes
0answers
65 views

What is the unitary $1$-parameter group generated by a vector field on a manifold?

If $M$ is a (compact, Riemannian) manifold, and one complexifies its tangent bundle, is it possible to give a meaning to the "unitary group"-like expression $t \mapsto \exp (\mathrm i t X)$ when $X$ ...
6
votes
1answer
148 views

Reference Question: Boundary Value Problem for Dirac Operator on Manifold with a non-smooth boundary

I am trying to find references for the following boundary value problem: Assume that $\Omega$ is a compact 3-dim spin manifold with Dirac operator $D$ such that the boundary consists of two smooth ...
1
vote
2answers
129 views

The existence of length-minimizing path between two points in a Riemannian manifold with boundary

Let $(M^n,g)$ be a Riemannian manifold with non-empty smooth boundary $\partial M$. For any two points $x,y\in M$, the distance between $x$ and $y$ may be defined as $$ d(x,y)=\inf_\gamma Length(\...
6
votes
0answers
67 views

Counter-examples to the higher dimensional statement of the half-space theorem

The well-known Half-space Theorem by Hoffman and Meeks says that there is no nonflat complete properly embedded minimal surface contained in an half space of $\mathbb{R}^3$. The higher dimensional ...
0
votes
1answer
62 views

A sufficient condition for isometrically embedding of manifolds in the Euclidean space they have already sat

Assume that $M$ is a submanifold of $\mathbb{R}^n$ and is equipped with a Riemannian metric such that the parallel transports associated with corresponding LC conection preserve the inner products of ...
1
vote
0answers
37 views

Does a map which preserve harmonic forms preserve co-closed forms (locally)?

$\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$ Let $\M,\N$ be $d$-dimensional oriented Riemannian manifolds ($d \ge 2$). Let $f:\M \to \N$ be smooth. Let $1 \le k \le d-1$ be fixed....
0
votes
0answers
42 views

Euler-Lagrange Equation for system with holonomic constraints

I have a very basic question, but i am not sure, that i understand the formula correctly. I have the Lagrangian for a many particle system. The coordinate of the particles build a N-dimensional ...
3
votes
1answer
93 views

Atoric equation

I'm looking for a general equation/function z = f(x, y, radius1, radius2, p1, p2) for an atoric surface. p1 and p2 could be either eccentricity or conic constant values. Can anyone help me with that? ...
3
votes
1answer
196 views

Slice theorem for proper groupoids

Let $G$ be a locally compact Hausdorff (second countable) groupoid with Hausdorff (second countable) unit space $X$. Assume $G$ is étale, i.e., the source and range maps of $G$ are local ...
3
votes
0answers
93 views

$\partial \overline{\partial}$-lemma for Irreducible, Normal Projective Varieties

Reference: W. Ding, G. Tian -- Kähler--Einstein metrics and the Generalised Futaki Invariant, Inventiones mathematicae, (1992). Let $X$ be a normal projective variety which is irreducible. Given an ...
1
vote
0answers
49 views

sequence definition of proper group action

My understanding is that for an action by a Lie group $G$ on a second countable and Hausdorff differentiable manifold $M$ to be proper, it suffices to show that the map $G \times M \rightarrow M \...
8
votes
1answer
126 views

Unique factorisation of prime geodesics?

In T. Sunada's 1985 paper ``Riemannian coverings and isospectral manifolds'', it is noted that for a compact Riemannian manifold $M$, prime (closed and non self-intersecting) geodesics behave like ...
0
votes
0answers
70 views

Lifting Levi Civita connection

When in differential geometry one shows , on a riemannian manifold, that a (unique) connection exists, (Levi Civita connection), is it possible to "lift" that notion to the principal bundle of frames ...
4
votes
0answers
59 views

Can we define a normal vector field on the level sets of the distance function?

Suppose $M$ is a smooth connected complete Riemannian manifold of dimension $n\geq 2$. Let $d:M\times M\rightarrow \mathbb{R}^+$ be the distance induced by the Riemannian metric on $M$. For $p\in M$ ...
-1
votes
1answer
74 views

A non-trivial upper bound on the integral of Lipschitz functions over a bounded support

Let $x \in \mathcal{X} = [0,1]^n$, and $f(x)$ be an $L$-Lipschitz function. Let $f(0)=0$. What is (the exact or a non-trivial upper bound on) $\int_{x\in\mathcal{X}} |f(x)|\,\mathrm{d}x$?What about $\...
4
votes
0answers
68 views

Representation on square integrable sections of a principal bundle

Let $X\rightarrow Y$ be a smooth principal $G$-bundle for some Lie group $G$. Then $L^2(X)$ has a natural $G$-action determined by fibrewise action of $G$ on $X$. We have an abstract isomorphism of ...
-1
votes
0answers
48 views

What is $H_{3}Spin(3)$, and how is this related with the twist of framing on a 3-manifold? [migrated]

From the question https://www.physicsoverflow.org/32208, Mr Ryan Thorngren said in the answer that the the framing anomaly of the gravitational Chern-Simons action $$I(g)=\frac{1}{4\pi}\int_{M}\...
3
votes
0answers
69 views

Verification of Gauss Bonnet theorem in vicinity of pseudosphere cuspidal geodesic equator $K=-1$

In the Gauss-Bonnet theorem with Euler characteristic $\chi=1$ $$ \int k_gds + \int K dA + \Sigma \psi_i = 2 \pi \tag1 $$ let us consider a triangle bounded by three geodesics on a constant $K=-1$ ...
1
vote
2answers
63 views

Dirichlet problem for capillary equation over convex domain

Let $\Omega \subseteq \mathbb{R}^2$ be a bounded convex domain with piecewise smooth boundary. Let $\phi :\partial \Omega \to \mathbb R$ be a continuous function. Let $L$ be a quasilinear elliptic ...
6
votes
2answers
326 views

A non integrable distribution which is totally geodesic

Is there a non integrable $2$ dimensional distribution $D$ of a $3$ dimensional Riemannian manifold such that the distribution is totally geodesic in the following sense: Every geodesic whose ...
9
votes
1answer
228 views

Current vs Varifold

I know the basic definitions concerning current and varifold, and they are generalization of submanifolds. What are their respective pros and cons? What are their crucial similarities and differences?
3
votes
1answer
85 views

Smoothness of a curve vs. smoothness of the squared distance from the curve to points on Riemann manifolds

I know that the squared distance function from a point $p$ on a Riemann manifold $M$ is smooth in a n-hood of $p$. Therefore for a smooth curve $c:\mathbb{R}\to M$ the concatenation $d(p,\cdot)^2\circ ...
0
votes
1answer
65 views

Rotation invariant of surface

Let $(x, y, f(x,y))$ be a surface in $\mathbb{R^3}$. It is written in a book without proof that all rotation invariant (rotating around $z$-axis) of $f$ are combinations of the following four ...
3
votes
2answers
145 views

Is the development map in Hyperbolic geometry related to development in Cartan geometry?

I am more familiar with Cartan geometry, and in this setting we have a notion of development of curves. As described in Cap & Slovak 1.5.17, on a Cartan geometry $(\mathcal{P} \to M, \omega)$ ...
4
votes
0answers
165 views

Pullback of Morse form satisfies Palais Smale

Let $(\alpha,g)$ be a Morse-Smale pair on a closed smooth manifold $M$, i.e. $\alpha$ is a Morse form and $g$ a Riemannian metric on $M$ such that stable and unstable manifolds of the gradient vector ...
16
votes
2answers
412 views

What is the weakest negative curvature condition ensuring a manifold is a $K(G,1)$?

The only statement I'm sure of is that any hyperbolic or Euclidean manifold is a $K(G,1)$ (i.e. its higher homotopy groups vanish), since its universal cover must be $\mathbb H^n$ or $\mathbb E^n$. ...
4
votes
0answers
229 views

A cohomology associated to a symplectic manifold

Let $(M,\omega)$ be a symplectic manifold. Let $$\Omega_{\omega}^k(M)=\{\alpha \in \Omega^{k}(M)\mid \alpha \wedge \omega \;\;\text{is an exact form}\}$$ Then we have a chain comlex$$\...