Questions tagged [dg.differential-geometry]
Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.
6,532
questions
1
vote
0answers
17 views
conformal changes to Lorentzian curvature
Let $(M,g)$ be a Lorentzian manifold and let $R$ be the curvature tensor. We say $R\leq 0$ if
$$ g(R(X,Y)Y,X) \leq 0\quad \forall \, X,Y \in TM.$$
My question is whether given a Lorentzian manifold $...
2
votes
1answer
145 views
Is the intersection of two distinct sufficiently small metric spheres always empty, a point or a metric sphere of lower dimension?
Let $(X,d)$ be an $n$-dimensional $(n< \infty)$ complete geodesic metric space, where any two points in $X$ are joined by a unique shortest geodesic. Let $S$ be a sufficiently small metric $(n-1)$-...
2
votes
0answers
38 views
Derive how the level quantization for 3d quantum Chern-Simons theory path integrals?
Let us consider abelian and non-abelian 3d quantum Chern-Simons theory path integrals:
abelian Chern-Simons theory on non-spin manifolds ---
$$
\int [DA]\exp(i \frac{k}{2\pi} \int_X (A \wedge dA ))
...
5
votes
1answer
78 views
Scalar product of random unit vectors
Let $X,X'$ be two random vectors on the sphere $S^{d-1}$. What is the distribution of their dot product $X\cdot X'$ in the following cases:
$X,X'$ independent with uniform distribution on the sphere $...
-2
votes
0answers
41 views
Compute the connection matrix
The (pseudo)Riemannian metric and the connection in $R^3$ with coordinates $x=u_1$,$y=u_2$,$z=u_3$ are given on the basic vector fields by
$$(\partial_{u_i}, \partial_{u_j})=\frac{\partial^3f}{\...
0
votes
0answers
46 views
Characterizing Lagrangian submanifolds of odd-symplectic manifold
Theorem 4.57 & 4.62 of Mnev's paper BV formalism and applications state the following:
Theorem 4.57 (ii) in Mnev's paper
Let $(\mathcal M, \omega)$ be an odd-symplectic manifold with body $...
-1
votes
1answer
95 views
Anti-symmetric operators for the Dirac or Majorana spinors
In a Zoom lecture given by a mathematical physics professor, if I recalled correctly, he explained that the in 1+1 dimensional spacetime (or 2 dimensions in short), the "action" of fermions (spinors) ...
2
votes
0answers
62 views
Equivariant resolution of singularities with equivariant centres
From what I understand, given a complex projective variety X inside a compact complex manifold Y, according to Hironaka, there is a sequence of $r$ blowups $Y_i$ of Y along complex submanifolds (...
6
votes
1answer
136 views
Cobordism monopole Floer homology
From the famous book: Monopole and three manifold, Kronheimer and Mrowka(https://www.maths.ed.ac.uk/~v1ranick/papers/kronmrowka.pdf). It is known that:
Let $Y$ be a closed oriented $3$ manifold, ...
-2
votes
1answer
231 views
Is the 2-dimensional Gauss-Bonnet theorem applicable in higher dimensions? [closed]
This is a cross-post of this MSE post that users commented that it is appropriate for MO.
I want to know
Question: Is the 2-dimensional Gauss-Bonnet theorem applicable (any topological ...
7
votes
1answer
302 views
+50
Size issues (small/large categories) when defining stacks in the Algebraic/differentiable/topological setting
Angelo Vistoli in the notes Notes on Grothendieck topologies, fibered categories and descent theory starts the section of category theory with the following note:
We will not distinguish between ...
6
votes
2answers
558 views
Curvature of nonsymmetric metric tensors?
Consider a smooth manifold $M$ of arbitrary dimension. We have notions of psuedo-Riemannian or Riemannian metrics on a manifold, and they differ in the slightest way of being positive-definite or not. ...
-2
votes
0answers
60 views
The area of the strip [closed]
Let $\gamma$ be a convex smooth plane curve of length $l$.
I need to compute the area of the strip swept by the outer normal segments of length $r$ to $\gamma$ and the length of the outer boundary ...
3
votes
0answers
76 views
Stokes's Theorem with singularities on projective line
Let $X$ be a complex manifold and $\omega\in \Omega(X\times \mathbb{P}^1)$ a form. I met the following identity:
$$\int_{\mathbb{P}^1}(\partial_z\bar{\partial}_z\omega) \log|z|^2=\int_{\mathbb{P}^1}\...
1
vote
0answers
56 views
Levi-Civita connection from idempotents
Let $(M,g)$ be a closed Riemannian manifold. Let $V$ be a smooth complex vector bundle over $M$. We can write $V$ as the range of an idempotent $E$ in a matrix algebra $M_n(C^\infty(M))$ acting on a ...
5
votes
0answers
102 views
An inequality for the Hessian of eigenfunctions of the Laplacian on compact manifolds
Let $(M,g)$ be a compact Riemannian manifold, and let $\Delta$ be the Laplace-Beltrami operator. Let $\lambda_1 >0$ be the first positive eigenvalue. That is, there exists a non-trivial function ...
-2
votes
0answers
49 views
where is the condition class $C^{1}$ used in the definition of current of integration? [closed]
Here is the usual definition of current of integration.
Let $Z \subset M$ be a closed oriented submanifold of $M$ of dimension $p$ and class $C^{1} ; Z$ may have a boundary $\partial Z$. The ...
2
votes
0answers
33 views
Lorentzian manifolds of negative spacelike sectional curvature
Suppose $(M,g)$ is a Lorentzian manifold of signature $(-,+,\ldots,+)$ and such that the sectional curvature of any space-like surface is non-positive. Is it true that there are no conjugate points or ...
6
votes
1answer
426 views
Which books should I read in order to be prepared to study information geometry?
At the moment, I am preparing my master's thesis (in statistics) and I intend to keep studying in order to pursue a doctoral degree. To be precise, I am mainly interested in studying Information ...
0
votes
0answers
44 views
Relation between subspaces of diagonal matrix and its “sign” matrix
Let $D$ be a $n \times n$ diagonal matrix with both positive and negative (but all non-zero) entries. Let $J = sign(D)$ be the matrix of $1$s and $-1$s representing the signs of the entries of $D$. ...
3
votes
0answers
137 views
Cobordism theory of some weird space
Let $G=SU(3)$ and $N=SO(3)$, then $G/N= SU(3)/SO(3)$ = a 5-dimensional Wu manifold $W$.
The $W$ is a homogeneous space (also a quotient space), but not a group.
Previously, I am aware of the ...
0
votes
0answers
16 views
Distance function and Hessian in Lorentzian geometries with positive curvature
Suppose $(M,g)$ is a Lorentzian manifold with signature $(-,+,\ldots,+)$ and a positive curvature. Let $p \in M$. Let $U$ be a sufficiently small neighborhood of $p$ in the exterior of the double null ...
6
votes
1answer
197 views
An extension of symplectomorphism group
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Sp{Sp}$Let $\omega=\sum dx_i\wedge dy_i$ be the standard symplectic structure of $\mathbb{R}^{2n}=\mathbb{R}^{n}\times \mathbb{R}^n$.
We consider the ...
1
vote
0answers
42 views
On smooth extensions of functions
Let $f(x) = \left(I - \hat{n}\hat{n}^T \cdot\textbf{1}_{\vec{n}^TAx \geq 0}\right)Ax$, where $I$ is the identity matrix, $A$ is a (symmetric) $d\times d$ positive definite matrix, $\hat{n}$ is an ...
6
votes
1answer
144 views
Mean curvature flow and knot theory
I am wondering if the mean curvature flow of one-dimensional submanifolds of $\mathbb{R}^3$ is understood well enough to give some perspective on (and hopefully a proof of) something like the Fary-...
2
votes
1answer
74 views
Extension of outer unit normal vector to interior
Suppose we have a bounded smooth domain $\Omega$ in $\mathbb{R}^n$, so there exists an outer unit normal vector field $\eta$ everywhere on the boundary. Can we extend it to the interior satisfying ...
6
votes
0answers
94 views
Examples of connection preserving maps in differential geometry
In synthetic differential geometry and tangent categories, linear connections on the tangent bundle are treated as a sort of algebraic gadgets that incorporate the tangent bundle. Like any other ...
2
votes
0answers
44 views
When is the quotient of a manifold by a discrete group of diffeomorphisms a diffeological covering space?
I was reading An Introduction to Diffeology by Patrick Iglesias-Zemmour and he defines a diffeological covering space as a diffeological fiber bundle with discrete fiber.
My question: Consider a ...
57
votes
28answers
5k views
Proofs where higher dimension or cardinality actually enabled much simpler proof?
I am very interested in proofs that become shorter and simpler by going to higher dimension in $\mathbb R^n$, or higher cardinality. By "higher" I mean that the proof is using higher dimension or ...
1
vote
1answer
125 views
Question about Berezin line bundle of odd cotangent bundle of supermanifold $\text{Ber}(\Pi T^*\mathcal N)$
The followings are from Mnev's paper about BV formalism.
Example 4.15 (Definition of split supermanifold)
Let $E \to M$ be a rank $m$ vector bundle over $n$-manifold $M$, then there exists a ...
6
votes
1answer
109 views
Notions of Lie 2-groupoids
The term Lie $2$-groupoid is used in the literature in more than one context. Some examples are given below:
Ginot and Stiénon's paper $G$-gerbes, principal $2$-group bundles and characteristic ...
6
votes
0answers
86 views
Length and curvature for closed curves in negatively curved spaces
In the Euclidean plane, for a closed smooth curve of length $\ell$ whose curvature is bounded above by $\epsilon$ we have the inequality
$$ \ell \ge 2\pi \epsilon^{-1} $$
which follows from the fact ...
2
votes
0answers
42 views
Closed-form expression for Riemannian exponential maps on symmetric spaces
Besides the Poincaré model for the hyperbolic disc, $S^n$, and Euclidean space, what are known instances of a symmetric space $M$ (finite-dimensional) for which the exponential map is known in closed-...
1
vote
0answers
109 views
When a free action gives rise to a $G$-principal bundle
When a free action gives rise to a $G$-principal bundle
Let a (topological) group $G$ act freely on a (topological) space $X$. Assume that
$G \backslash X$ is Hausdorff. (equivalently the image of ...
3
votes
0answers
54 views
Commutation relations between covariant and Lie derivatives
I am currently working on extrinsic riemannian geometry and I am looking for a sort of commutation relation between the covariant and Lie derivatives.
To be more precise : considering an hypersurface ...
3
votes
1answer
72 views
Tangent bundle with Sasaki metric is Kähler iff $M$ is locally flat
I'm having a hard time proving the following:
If $M$ is an $n$-dimensional indefinite Riemannian manifold whose metric $g$ has index $s$, then the metric of Sasaki $g^{D}$ is an indefinite metric ...
0
votes
0answers
34 views
Difference of two functions with constant mean curvature
Define the set $\Omega := (-\epsilon,\epsilon) \times (-1,1)^{n-1}$, and define
$\Gamma := \{-\epsilon,\epsilon\} \times (-1,1)^{n-1} \subset \partial \Omega$.
Suppose I have two functions $u,v \in C^...
1
vote
1answer
57 views
Initial value problems on manifolds around submanifolds (reference)
I am looking for a reference on initial value problems formulated on smooth manifolds with initial conditions on submanifolds. More precisely, let $X$ be a smooth manifold and $Y\subset X$ a embedded ...
1
vote
0answers
54 views
On submodules of vector fields
I don't know much about modules aside from their basic definition and that they are more complicated than vector spaces.
I am asking this question because I wish to have a more "algebraic" ...
2
votes
1answer
141 views
A metric naturally arise from the Euclidean symplectic structure?
For $n>1$ let $\omega=\sum_{i=1}^n dx_i\wedge dy_i$ be the standard symplectic structure on $\mathbb{R}^{2n}=\mathbb{R}^n \times \mathbb{R}^n$.
We define the following distribution $D$ on $\mathbb{...
3
votes
1answer
85 views
About the metric and embedding of sphere
Let $S^2$ be the $2$-dimensional sphere with a metric $g$.
Q:
Can we or how to find a smooth map $f:S^2\to \mathbb R^3$, such that
(1) $f$ is diffeomorphic to its image $Im(g)=:M$,
(2) $M$ ...
0
votes
0answers
23 views
$C^l$-extension preserving Hölder continuity of derivatives
Let $b : D\times [0,T] \to \mathbb R^d$ be a function with $D\subseteq \mathbb R^d$ closed, $T > 0$ and suppose $b$ is in $C^l(\operatorname{int} D\times (0,T))$.
Perhaps then we could extend $b$ ...
7
votes
2answers
252 views
Vector field with constant divergence around embedded submanifold
Let $M$ be a smooth $n$-dimensional manifold and $N\subset M$ be a closed embedded submanifold of codimension at least $2$. Furthermore, let $\mu$ be a volume form on $M$.
Question: Does there ...
0
votes
0answers
36 views
prove a bondle is an indefinite Hermitian manifold which is Kahler if and only if the manifold is locally flat
Let $M(J,g)$ be an indefinite Kahler manifold, then $%
TM(J^{H},g^{D})$ is an indefinite Hermitian manifold which is Kahler if and
only if $M$ is locally flat. Here $J^{H}$ denotes the horizontallift ...
8
votes
0answers
127 views
Reference request: Name or use of this group of diffeomorphisms of the disc
Let $k \in \{0,\infty\}$, $G\subseteq \operatorname{Diff}^k(D^n)$ be the set of diffeomorphisms $\phi:D^n\to D^n$ of the closed $n$-disc $D^n$ (with its boundary) satisfying the following:
$
\phi(S_r^...
0
votes
0answers
27 views
proof of the following theorm on simply-connected, complete indefinite Kahler manifold
can anyone help me prove the following therorm
If $c\in \mathrm{I\!R}$ every connected, simply-connected, complete
indefinite Kahler manifold of complex dimension $n$, of index 2s and of
constant ...
7
votes
0answers
139 views
Which convex bodies can be captured in a knot?
Which convex bodies can be captured in a knot?
This question is based on the discussion in "Is it possible to capture a sphere in a knot?".
We assume that the knot is made from unstretchable, ...
2
votes
1answer
126 views
$\mathbb Z$-graded manifold is isomorphic to the structure sheaf of supermanifold locally
In this paper, definition 4.4.1 about supermanifold and definition 4.6.1 about graded manifold:
Definition 4.4.1: An supermanifold $\mathcal{M}$ is a locally ringed space $(M,\mathcal O_M)$ which ...
1
vote
0answers
53 views
zero extension of positive currents are always positive
In Demailly's Complex Analytic and Differential Geometry page 139:
He said the trivial (zero) extension of the positive current $T$ (on $X\setminus E$), which denoted by $\tilde T$ is always positive ...
1
vote
0answers
96 views
Existence theory with an integral equation
I am reading a paper in which it is proposed that one can solve a problem from mathematical physics by establishing an existence theory for a system of equations. One of the equations in the system ...
