All Questions
Tagged with dg.differential-geometry smooth-manifolds
516 questions
4
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Integration of volume forms over manifolds with corners
Suppose that $M$ is a (compact, oriented, smooth) manifold with corners.
Let $M_{\leq 1}$ the manifold with boundary of all points of index $\leq 1$ (following the notations here). This manifold is an ...
- 1,035
-1
votes
1
answer
237
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Almost Complex Structure extending to Complex Structure, aka "Integrable"
Let $M$ be a smooth manifold of (real) dimension $2n$. An almost complex structure $J$ on $M$ is a linear vector bundle isomorphism $J \colon TM\to TM$ on the tangent bundle $TM$ such that $J^2 = − 1 \...
- 6,018
3
votes
0
answers
109
views
Wedge of curvature and subsequent trace
I am currently reading https://arxiv.org/abs/1901.10322. More specifically, I am interested in understanding the equation
$$i\partial\overline{\partial}\omega = \frac{\alpha'}{4}Tr(R\wedge R-F\wedge F)...
1
vote
0
answers
117
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Question on globally hyperbolic manifolds and coordinates
Consider a globally hyperbolic Lorentzian manifold $(M,g)$. Then, a well-known result of Bernal-Sánchez (see Theorem 1.1 in arXiv:gr-qc/0401112) states that it can globally be written as
$$M=\mathbb{R}...
- 1,171
5
votes
0
answers
122
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Explicit parallelization of an exotic sphere
I asked this question on MathStackExchange a week ago (see here), but, despite a few upvotes, I received no comments or answers. Ideally, I would love a detailed answer, but a yes/no would do the job! ...
- 51
1
vote
1
answer
94
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Linear combinations of smooth sections
I am dealing with smoothness issues for which I do not even know a successful approach, so any help or reference would be welcome.
Let $M$ be a manifold, $E\to M$ a smooth vector bundle, and let $\...
- 1,452
3
votes
1
answer
189
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Randomly perturbed function has no accumulated critical point almost surely?
Given a smooth function $f$ and a smooth manifold $\mathcal{M}$ in $\mathbb{R}^d$, define the set
$$
S(v):=\{x:{\rm Proj}_{T_x{\mathcal{M}}}(v)=\nabla_{\mathcal{M}}f(x)\}.
$$
Is correct to say that $S(...
- 43
2
votes
0
answers
75
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Pullback by surjective submersion is injective?
Denote by $\mathcal{D}'_X$ the sheaf of distributions on a smooth manifold $X$.
Let $M$ and $N$ be smooth manifolds and $\Phi: M \to N$ a submersion. Then $\Phi$ defines a unique morphism of sheaves $\...
- 261
7
votes
1
answer
201
views
Lipschitz bounds and homotopy groups of diffeomorphism groups
Let $M$ denote a closed Riemannian manifold. Let $\mathrm{Diff}_0^L(M)$ denote the supspace of the identity component of the diffeomorphism group $\mathrm{Diff}_0(M)$ of diffeomorphisms with Lipschitz ...
- 1,174
4
votes
0
answers
134
views
Property of a smooth function is true for almost every function
Consider a smooth function $f:\mathbb{R}^n\to \mathbb{R}$. We further assume $0$ is a regular value of $f$, thus $X=f^{-1}(0)$ is a smooth submanifold of dimension $n-1$. Consider the following ...
- 71
5
votes
1
answer
393
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No analytic surjection $f:M \to N$ when $\dim(M) >\dim(N)$
Inspired by comment discussions in this MO post smooth version of splitting principle we ask:
Are there two compact real analytic manifolds $M,N$ of dimension $m>n$ such that there is not any ...
- 356
1
vote
0
answers
240
views
Smooth version of the splitting principle
Inspried by this MO question A manifold whose tangent space is a sum of line bundles and higher rank vector bundles we pose the following question as a possible smooth version of the splitting ...
- 356
2
votes
1
answer
122
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Slowly increasing smooth mappings with values in a Lie group?
Let $G$ be $l$-dimensional compact Lie group and consider any smooth $F : \mathbb{R}^n \to G$.
Then, the first-order derivative of $F$ at each $x \in \mathbb{R}^n$ can be regarded as a linear mapping $...
- 3,477
0
votes
1
answer
91
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Topological Properties of Subsets of $R^{m}$ induced by Smooth Manifolds in Matrix Spaces
We know that $M_{m \times n } $ is isomorphic to $R^{mn}$. Let's take a smooth manifold $\mathbb{M}$ in $R^{mn}$ and fix a point in $R^{n}$, say, $p$. Now realize the manifold $\mathbb{M}$ as a subset ...
- 101
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0
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80
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Details of the proof of the inequality $ \int_{X}\left(2 r \mathrm{c}_{2}(E)-(r-1) \mathrm{c}_{1}^{2}(E)\right) \wedge \omega^{n-2} \geq 0.$
I'm trying to make sense of the following proof.
Let $E$ be a holomorphic vector bundle of rank $r$ on a compact hermitian manifold $(X, g)$. If $E$ admits an Hermite-Einstein structure then $$
\int_{...
- 103
2
votes
0
answers
107
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Finite dimensional manifolds as subspace of $\mathbb{R}^\mathbb{N}$
For embedded submanifold, specifically with ambient space being $\mathbb{R}^{n}$, there are many nice properties and results. Specifically there are many examples of matrix manifolds such as the ...
- 275
2
votes
0
answers
101
views
A roof genus of high dimensional lens space
Let $p$ be a natural number, and for $i\in \{0,
..., p-1\}$,
denote the irreducible rank one complex representation of $\mathbb{Z}/p$. by $\rho_{i}$.
Let $a=(a_{1},\ldots a_{d}) $ ...
- 2,630
3
votes
1
answer
286
views
Unitary transformations of Vandermonde matrices forms a smooth manifold?
The space of all Vandermonde matrices $V$ with $r$ variables and degree $n$ (as below) forms an embedded submanifold of $\mathbb{R}^{(n+1) \times r}$ when $x_{i} \in \mathbb{R}$. It is naturally a ...
- 275
8
votes
1
answer
230
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The closure of the space of Riemannian metrics with a fixed isometry class
Let $M$ be a closed manifold, and let $\mathscr{M}$ be the space of all Riemannian metrics over $M$. It is known that this is a Fréchet manifold. Consider also $\mathscr{D}$ the diffeomorphisms group ...
- 808
6
votes
1
answer
272
views
Ideals of functions whose zero locus is a submanifold
Let $M$ be a smooth $m$ dimensional manifold. Suppose that $f_1,\dots,f_k\in C^\infty(M)$ are smooth functions such that the zero locus $$N:=Z_f=\lbrace p\in M:\ f_i(p)=0,\ \forall i=1,2,\dots,k\...
- 2,792
1
vote
1
answer
182
views
Lower bound on injectivity radius at one point implies lower bound on injectivity radius for a closed manifold
I’m interested in a closed Riemannian manifold $(M^n,g)$ with $sec<0$ and $diam(M)\leq D$. My question is:
If at some point $p\in M^n$, the injectivity radius $injrad(p)\geq1$, then can we get $...
- 155
3
votes
2
answers
254
views
First examples of Lie-Rinehart algebras that are not coming from Lie algebroids
I heard the idea of a Lie-Rinehart algebra first time from an algebraist.
I noticed there is a similarity between description of Lie algebroid on a manifold and the algebraic notion of Lie-Rinehart ...
- 6,023
6
votes
0
answers
127
views
Is there a canonical smooth structure on tame Fréchet orbit type stratifications?
In finite dimension orbit type stratifications, it is known that the orbit space $M/G$ resulting from an action of a proper Lie Group $G$ on a smooth manifold $M$, satisfying a set of certain ...
- 808
0
votes
0
answers
59
views
Condition to show $\{ U \in \mathbb{R}^{n \times p}|\mathscr{A}(UU^{\top}) = b \}$ is (is not) a manifold
Consider $\mathscr{A}: S^{n\times n} \to \mathbb{R}^{m}$, $b \in \mathbb{R}^{m}$, I would like to know when $\mathscr{M}:=\{ U \in \mathbb{R}^{n \times p}|\mathscr{A}(UU^{\top}) = b \}$ is a manifold. ...
- 275
5
votes
3
answers
1k
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Naturality of Lie bracket — alternate proof
Let $M$ and $N$ be smooth manifolds, and let $F: M \to N$ be a smooth map. Let $X$ and $Y$ be vector fields on $M$, and let $\tilde{X}$ and $\tilde{Y}$ be vector fields on $N$. We say that $X$ and $\...
- 807
2
votes
0
answers
46
views
Under what conditions principal directions define an integrable distribution?
Consider a hypersurface $M^n \subset \mathbb{R}^{n+1}$ which is compact without boundary. Assume that its second fundamental form $A$ has distinct eigenvalues $\lambda_1<\ldots<\lambda_k$ (with $...
- 363
1
vote
0
answers
88
views
Metric of negative curvature on connected sum
Let $(M_1,g_1)$ and $(M_2,g_2)$ be two Riemannian manifolds of dimension $n\geq 2$. If we consider the connected sum $M=M_1\mathbin{\#}M_2$ of the two manifolds; can one get a smooth metric $g$ on $M$ ...
- 11
5
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0
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337
views
Milnor’s smoothed corners technique for a product of manifolds with boundary and boundary defining functions
Let $M$ be a smooth manifold that we view as the interior of a compact manifold with boundary $\overline{M}$. Let $\rho$ be a boundary defining function for $\overline{M}$, i.e. $\rho$ is smooth, $\...
- 319
-4
votes
1
answer
328
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Does a coarser topology lead to a non-Hausdorff topological manifold? [closed]
Take a topological manifold $M$. Suppose one considers a strictly coarser topology than the manifold topology. Can such topology result in a non-Hausdorff topological manifold?
NOTE: PLEASE avoid the ...
- 612
3
votes
1
answer
329
views
What is known about the "stickiness" of a smooth manifold to its tangent space?
Consider an $m$-dimensional smooth manifold $M$ embedded in $n$-dimensional real Euclidean space $\mathbb{R}^n$ ($n>m$). Consider a point $x \in M$ of the manifold, and for simplicity, choose the ...
- 31
10
votes
1
answer
608
views
Disintegration measures and differential forms
Let $X$ and $Y$ be smooth oriented manifolds of dimension $m$ and $n$, and let $f\colon X\to Y$ a proper smooth map. There is a theorem called the "Disintegration Theorem" which says ...
- 44.7k
5
votes
1
answer
343
views
Clarifying a result of Klingenberg
I am looking for some clarification on a result of Klingenberg. For context, I have a complete Riemannian manifold for which I would like to compute a lower bound on the injectivity radius at each ...
- 163
0
votes
1
answer
202
views
How to estimate the distance between geodesics and points for Riemannian manifold with positive sectional curvature
Assume that $ M $ is a complete Riemannian manifold and there exists $ k>0 $ such that $ K(q)\geq k $ for any $ q\in M $, where $ K $ is the sectional curvature of $ M $. Let $ \gamma $ be a closed ...
- 1,219
6
votes
1
answer
273
views
Commutative/ symmetric second covariant derivative
Consider a smooth manifold $M$ together with an affine connection (or covariant derivative) $\nabla$ on the tangent bundle $TM$.
Is it possible to have an affine connection, possibly with non-zero ...
- 61
4
votes
2
answers
583
views
Compactification of a product of manifolds
Let $M$ be a smooth manifold. We make the assumption that $M$ can be viewed as the interior of a compact manifold with boundary $\overline{M}$. In practice, for an explicit manifold, any ...
- 319
14
votes
2
answers
2k
views
Is a manifold Euclidean if its tangent bundle is Euclidean?
I'm wondering whether an $n$-dimensional manifold diffeomorphic to $\mathbb{R}^n$ if its tangent bundle is diffeomorphic to $\mathbb{R}^{2n}$. Many thanks!
- 149
2
votes
1
answer
104
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Why is this subset associated to a $2$-tensor dense?
Let $S$ be a symmetric $(0, 2)$ tensor on a Riemannian manifold $M$. Define $E_S : M \to \mathbb{Z}$ by $E_S(x) = \left(\text{the number of distinct eigenvalues of } S_x\right)$. I've seen the ...
- 659
1
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0
answers
247
views
Cokernel of the Jacobian Matrix
For an algebraic variety $V = \mathcal{V}(f_1,\dots,f_m)\subset \mathbb{C}^n$, in smooth points $p$ there is a nice geometric interpretation of the Jacobian $(\partial f_i/\partial x_j)_{ij}\lvert_p$'...
2
votes
0
answers
132
views
Elliptic equations and Fredholms alternative in the non-compact case
Let $M$ be a smooth Riemannian manifold and $E$ be a finite-rank vector bundle over $M$ equipped with a bundle metric $\langle\cdot,\cdot\rangle\in\Gamma^{\infty}(E^{\ast}\otimes E^{\ast})$, i.e. $\...
- 1,429
5
votes
1
answer
311
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Diagonalization of symmetric matrices of functions
I asked this question some time ago in MSE but I didn't recieved any feedback.
https://math.stackexchange.com/questions/4672664/diagonalization-of-symmetric-matrices-of-functions
This problem arised ...
- 243
8
votes
2
answers
537
views
Compactly-supported harmonic tensors
Let $({M},g)$ be a connected and non-compact Riemannian manifold without boundary. If $L:\Gamma^{\infty}(E)\to \Gamma^{\infty}(E)$ is a linear second order elliptic operator on some smooth $\mathbb{R}$...
- 1,171
6
votes
1
answer
423
views
Difference between parallel transport and ambient projection
Consider a $d$-dimensional complete embedded Riemannian submanifold $(M,g)$ of a Euclidean space $\mathbb{R}^D$ (The major examples we consider are sphere and Stiefel manifold). Assume the sectional ...
- 125
2
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0
answers
182
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Proving the canonical form of a vector field near a regular point on the boundary [closed]
I'm trying to solve Theorem 9.35 of Lee's Introduction to smooth manifolds:
Let $M$ be a smooth manifold with boundary and let $V$ be a smooth vector field on $M$
that is tangent to $\partial M$. If $...
- 21
5
votes
1
answer
1k
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Basic question on the de Rham theorem
There is a modern nice proof of the de Rham theorem based on sheaf theory.
The de Rham theorem says that for a smooth manifold $M$ there is a canonical isomorphism
$$H^i_{dR}(M,\mathbb{R})\simeq H^i_{...
- 21.8k
4
votes
0
answers
334
views
Hodge decomposition on non-compact manifolds
Let $(\mathcal{M},g)$ be a compact Riemannian manifold without boundary. Then we have the well-known Hodge decomposition
$$\Omega^{k}(\mathcal{M})\cong\mathcal{H}^{k}(\mathcal{M})\oplus\mathrm{ran}(\...
- 1,171
4
votes
1
answer
212
views
Existence of eigen basis for elliptic operator on compact manifold
Let $M$ be a compact Riemannian manifold. Let $E$ be a vector bundle over $M$ equipped with a Hermitian (or Euclidean) metric on its fibers. Let $D$ be a linear elliptic differential operator acting ...
- 21.8k
10
votes
1
answer
849
views
Hodge decomposition in elliptic complexes
EDIT: In the book "Principles of Algebraic Geometry" by Griffiths and Harris the authors prove the Hodge decomposition for the Dolbeault operator $\bar\partial$ on differential forms on a ...
- 21.8k
5
votes
3
answers
620
views
Poisson equation on manifolds
Let $(\mathcal{M},g)$ be a compact Riemannian manifold with Levi-Civita connection $\nabla$. It is well-known that the Poisson equation
$$\Delta u=f$$
does have a solution on $C^{\infty}(\mathcal{M})$ ...
- 1,171
8
votes
2
answers
454
views
Two different spin structures of the real projective space $\Bbb RP^3$
It is known that every orientable 3-manifold has a spin structure, because its tangent bundle is trivial. Also it is known that if a manifold $X$ has a spin structure, then the number of distinct spin ...
- 335
2
votes
1
answer
342
views
Sufficient condition for the union of two submanifolds to be a submanifold
I have two smoothly embedded orientable surfaces $S_1,S_2\subset S^3 \times [0,1]$ with boundary such that
$(i)$ $S_1\cap S_2$ is a smoothly embedded surface without boundary and
$(ii)$ $\overline{...
