Questions tagged [smooth-manifolds]
Smooth manifolds and smooth functions between them. For manifolds with additional structure, see more specific tags, such as [riemannian-geometry]. For more topological aspects, see [differential-topology].
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Total Mean Curvature as a integral on the whole space
It is well known from De Giorgi that we may express the surface area of a domain $\Omega\subset\mathbb{R}^N$ as:
$$
\int_{\partial\Omega} 1\ d\sigma=\int_{\Omega} ||\nabla H(\phi(x))||\ dx=\int_{\...
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Spacetime symmetries
We know some nice space-time have a lot of symmetries. It is said that
Minkowski spacetime has
$$ISO(d-1,1)/SO(d-1,1),$$
de Sitter spacetime has
$$SO(d,1)/SO(d-1,1)$$ and
anti-de Sitter spacetime ...
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Example metrics for exotic R4
I'm a physics student trying to understand what exotic manifolds, such as exotic R4, means. Is there known examples what the Riemannian metric of some exotic R4 (or some exotic sphere) would be? Does ...
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When do unknots in $S^n$ bound unique balls?
I recently heard the it is an open question whether or not an unknotted $S^2$ in $S^4$ bounds a unique $B^3$ in $S^4$, where by unique, I mean up to isotopy rel boundary in $S^4$. The rel boundary ...
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$\exists X$?(No basis for $H_2(X)$ contains spheres)
Does there exist a smooth simply connected closed 4-manifold $X$ with the property below?
Every smooth basis for $H_2(X)$ contains a surface with genus $\geq 1$.
I understand that in general the ...
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(Non-)Orientability of non-triangulable manifolds
We heard and learned from Mike Miller's answer to Not all manifolds can be triangulated: In which dimensions? that "All orientable 5-dimensional manifolds are triangulable. In dimensions at least ...
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Integrability/regularity of Lyapunov exponents
My question is about some basic properties of Lyapunov exponents. Sorry if this stuff is basic, I just don't know where to find the statements that I'm looking for.
Preliminaries. Let $X$ be a closed ...
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Inclusion of closed submanifolds of a manifold
Consider a smooth compact manifold $M$ of dimension $n$, with or without boundary. Choose a submanifold $N$ of $M$ of dimension $k$, where $1 \leq k \leq n - 1$, such that $N$ is either without ...
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What algebraic structure characterizes all natural operations between differential operators and differential forms?
On a smooth manifold $M$ one can define various algebraic structures, natural with respect to diffeomorphisms:
the differential graded-commutative algebra $\Omega(M)$ of differential forms on $M$;
...
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Look for a suitable cut-function: from Pierre Grisvard "Elliptic Problems in Nonsmooth Domains": (Theorem 1.4.2.4)
From Pierre Grisvard "Elliptic Problems in Nonsmooth Domains":
Theorem[Theorem 1.4.2.1]
Let $\Omega$ be an open subset of $\mathbb{R}^d$ with a continuous boundary, then $C_c^\infty(\...
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Real manifolds in a theorem prover?
Which of the formal computer proof verification systems (like Lean, Coq, Agda, Idris, Isabelle-HOL, HOL-Light, Mizar etc) have a basic theory of real manifolds? Up to, say, the definition of a smooth ...
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Geodesics (Local vs Global)
Let $M$ be a Riemannian manifold, and $p,q\in M$ two points. Now, if $M$ is a complete metric space the Hopf–Rinow theorem ensures that there is a geodesic $C\subset M$ joining $p$ and $q$ that ...
user75933
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Wildness of codimension 1 submanifolds of euclidean space
This question arose out of this stack exchange post. I am wirting a thesis about the $s$-cobordism theorem and Siebenmann's work about end obstructions. Combined they give a quick proof of the ...
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Reference request: A knot is tame if and only if it has a tubular neighbourhood
Definitions:
A knot is an embedding $\kappa:S^1\hookrightarrow S^3$ (we do not require smooth or polygonal).
Two knots $\kappa,\,\lambda:S^1\hookrightarrow S^3$ are equivalent if one of the following ...
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Immersions of surfaces in $\mathbb{R}^3$
Stephen Smale famously proved in [Trans. Amer. Math. Soc. 90 (1959), 281-290] that any two $C^2$ immersions $S^2\to\mathbb R^3$ are regularly homotopic. This is how we knew that one can do a sphere ...
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A result of Borel on extensions of arithmetic groups
A famous result of Sullivan (closely related to work of Wilkerson) says that the group of isotopy classes of diffeomorphisms of a simply-connected closed smooth manifold of dimension $\geq 5$ is ...
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Category of Manifolds and Maps: TOP $\supseteq$ TRI $\supseteq$ PL $\supseteq$ DIFF? [closed]
Please let me denote the following
(TOP) topological manifolds https://en.wikipedia.org/wiki/Topological_manifold
(PDIFF), for piecewise differentiable https://en.wikipedia.org/wiki/PDIFF
(PL) ...
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Novikov conjecture
The statement of the Novikov conjecture is a bit esoteric. Does the following simplified conjecture have any known counterexamples?
C: For a smooth closed 4n-fold $M$, the Pontryagin numbers are ...
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Is a smooth family of vector spaces always locally trivial?
Let's define a smooth family of vector spaces to be the following data $(E,M,p,a,s,z)$, where
(1) $p:E\to M$ is a smooth submersion of smooth manifolds
(2) There is some fixed integer $n\ge 0$ such ...
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Special Riemannian metric on the product
Let $M^2$ be an orientable surface with boundary endowed with a Riemannian metric $g$. We know that if we put in the manifold $M \times \mathbb{R}$ the product metric $\overline{g} = g + dt^2$, then ...
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"Smooth" Serre Fibrations (?)
Let $M,N$ be manifolds, $f:M \to N$ be a map.
In order to understand if $f$ is a serre fibration, it is enough to test it against differntiable maps $I^p \to M, I^{p+1} \to N$? What about smooth maps?...
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Asymptotic bound on minimum epsilon cover of arbitrary manifolds
Let $M \subset \mathbb{R}^d$ be a compact smooth $k$-dimensional manifold embedded in $\mathbb{R}^d$. Let $\mathcal{N}(\varepsilon)$ denote the minimal cardinal of an $\varepsilon$-cover $P$ of $M$; ...
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Subdividing a Compact Bounded Curvature Manifold into Charts with Bounded Lipschitz Constant
Let $M \subset \mathbb{R}^d$ be a compact smooth $k$-dimensional manifold embedded in $\mathbb{R}^d$. Let $\mathcal{N}(\epsilon)$ denote the size of the minimum $\epsilon$ cover $P$ of $M$; that is ...
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Is $π:\mathcal{C}^∞(M,N)→\mathcal{C}^∞(S,N)$, $π(f)=f|_S$ a quotient map in the $\mathcal{C}^1$ topology?
This question was previously posted on MSE.
Let $M, N$ be smooth connected manifolds (without boundary), where $M$ is a compact manifold, so we can put a topology in the space $\mathcal C^\infty(M, N)$...
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Symmetric and anti-symmetric parts of the covariant derivative of a connection
The following is an excerpt from Sharpe's Differential Geometry - Cartan's Generalization of Klein's Erlangen Program.
Now we come to the question of higher derivatives. As usual in modern
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Why do unstable manifolds of two close point intersect each other in Baker map?
Let $M$ be $S^1 \times [-1,1]$, $f$ a baker map on $M$ and for $p, q \in M$ consider $W^s_p$ the stable manifold in $p$ (i.e. the set of points whose forward orbit tend to the forward orbit of $p$) ...
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Euler-Lagrange equations on a differentiable manifold
I am following the conventions of https://arxiv.org/abs/math-ph/9902027
Let $M$ be a differentiable manifold, $E \to M$ a vector bundle over $M$ with fibre $F$, $J^1(E)$ the rank-one jet bundle over $...
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Does every large $\mathbb{R}^4$ embed in $\mathbb{R}^5$?
This question was prompted by my answer to this question.
An exotic $\mathbb{R}^4$ is a smooth manifold homeomorphic to $\mathbb{R}^4$ which is not diffeomorphic to $\mathbb{R}^4$ with its standard ...
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Local diffeomorphism on a neighborhood of an embedding
In my reading of the (excellent!) paper of Grabowski and Rotkiewicz on higher vector bundles (https://arxiv.org/abs/math/0702772), I have encountered the following argument which I do not understand. ...
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Volume comparison on Grassmannian
Let $G(r,n-r)$ be the Grassmannian, which can be identified with the space of all rank $r$ projection matrices in $\mathbb{R}^n$. Let $\mu$ be the uniform measure on $G(r,n-r)$. For any $\lambda_1,......
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Uniqueness of Fano varieties
It is a theorem of Kollár–Miyaoka–Mori that there is a finite number of deformation families of smooth, complex Fano $n$-folds for each $n$ (hence also a finite number of diffeomorphism types).
My ...
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Are normal coordinates the same as Cartesian coordinates in flat space? [closed]
Let $\gamma_v$ be the unique maximal geodesic with initial conditions $\gamma_v(0)=p$ and $\gamma_v'(0)=v$ then the exponential map is defined by
$$\exp_p(v)=\gamma_v(1)$$
If we pick any orthonormal ...
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Wu formula for manifolds with boundary
The classical Wu formula claims that if $M$ is a smooth closed $n$-manifold with fundamental class $z\in H_n(M;\mathbb{Z}_2)$, then the total Stiefel-Whitney class $w(M)$ is equal to $Sq(v)$, where $v=...
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Extending Green's theorem from very special regions to more general regions
Green's theorem
Let $C$ be a positively oriented and consists of a finite union of disjoint,piecewise smooth simple closed curve in a plane, and let $D$ be the region bounded by $C$. If $P$ and $Q$ ...
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Same fiber of induced covering map [closed]
Consider a holomorphic map $h: X \to E$ between compact, connected, complex analytic manifolds Let $p: \tilde{E}\to E$ be the universal cover, and denote by $\tilde{h}: \tilde{X}\to\tilde{E}$ the pull-...
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The idealizer of the space of vector fields with vanishing divergence
The space of smooth vector fields on a manifold is counted as a Lie algebra with the usual Lie bracket structure.
Is there a Riemannian manifold of dimension at least $2$ which satisfies either of ...
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Milnor immersion of circle, disks, and a ball
Milnor surprisingly found an immersion of a circle in the plane which bounds two "incompatible" immersed disks (see [1] for example, picture included). I think I can glue these two disks together to ...
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Integrating over some domain of the Stiefel manifold to analyze its support
Define the square $d$ dimensional Stiefel manifold as
$$V_{d} = \{ R \in \mathbb{R}^{d \times d} : R ^\top R = I_d \} .$$
How does one integrate on this manifold over a domain defined as $\{ R \in V_{...
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Existence of an isotopy in Riemannian manifold
Let $(M,g)$ be a Riemannian manifold, and $p,q\in M$ be two fixed points. We assume $p,q$ are close enough. Say, we assume $p$ and $q$ are in the same normal coordinate chart. It is clear that there ...
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Finite good covers on smooth manifolds
Let $M$ be a connected smooth manifold that is not necessarily compact but has the homotopy type of a finite CW complex.
Does $M$ admit a finite good cover? (i.e. a finite cover by contractible ...
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Infinitesimal matrix rotation towards orthogonality
TLDR; I am trying to prove the existence of an infinitesimal rotation which always moves a matrix "closer" to being orthogonal.
Setting
In this setting, we have a matrix $W \in \mathbb{R}^{n \times ...
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Extension of Vector Field in the $\mathcal{C}^r$ topology
This question was previously posted on MSE.
Let $M\subset \mathbb{R}^n$ be a compact smooth manifold embedded in $\mathbb{R}^n$, we define $$\mathfrak{X}(M) := \{X: M \to \mathbb{R}^n;\ X\mbox{ is ...
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Matrix trace minimization of quadratic and linear terms under orthogonal manifold constraints
How would one solve the following orthogonal manifold problem?
$$\begin{array}{ll} \text{maximize} & \mbox{tr}(X^\top A X - X^\top B)\\ \text{subject to} & X^\top X = I\end{array}$$
where $A ...
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Structural Stability on Compact $2$-Manifolds with Boundary
I'm studying the structural stability of vector fields and I'm interested in learning about this phenomenon on compact $2$-manifolds with boundary.
Let $M^2$ be a compact connected 2-manifold and $\...
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Smooth vector fields on a surface modulo diffeomorphisms
Let $\Sigma$ be a two-dimensional connected smooth manifold without boundary. (Feel free to assume it is compact and orientable.)
Let $\mathcal{X}(\Sigma)$ denote the smooth vector fields on $\Sigma$...
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Integration over a Surface without using Partition of Unity
Suppose we are given a compact Riemann surface $M$, an open cover $\mathscr{U}=\{U_1,U_2,\dots\}$ of $M$, charts $\{(U_1,\phi_1),(U_2,\phi_2),\dots\}$, holomorphic coordinates, $\phi_m:p\in U_m\mapsto ...
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On the homotopy type of $\mathrm{Diff}(\mathbb{S}^3)$
I am confused with the following argument. I know I am doing something wrong but I can't find my mistake.
On one hand, one knows that if $M$ is a Lie group, then
$$\mathrm{Diff}(M)\simeq M\times\...
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Traceless sobolev forms on compact manifolds with boundary
Let $(M,g)$ be a smooth, compact, oriented Riemannian manifold with smooth, oriented boundary $\partial M$. Further, let $\Omega^p(M)$ and $\Omega^p(\partial M)$ be the spaces of smooth differential $...
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Metrics on derived smooth manifolds
Derived geometry explains how to remove the transversality condition and make sense out of a nontransversal intersection.
For example, if $X$ and $Y$ are embedded submanifolds of a manifold (or ...
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History of the definition of smooth manifold with boundary
I am trying to determine the earliest source for the definition of smooth ($C^\infty$) manifold with boundary. Milnor and Stasheff (1958) give a definition, but a scrutiny of that definition shows it ...
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