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Questions tagged [submanifold]

In mathematics, a submanifold of a manifold $M$ is a subset $S$ which itself has the structure of a manifold, and for which the inclusion map $S \rightarrow M$ satisfies certain properties. There are different types of submanifolds depending on exactly which properties are required. Different authors often have different definitions. Therefore, please include the details when using this tag.

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Find $f$ such that $f^{-1}(\lbrace0\rbrace)$ is this knotted curve (M.W.Hirsh)

I would like to solve the following problem (it comes from Morris W. Hirsh, Differential Topology, it's exercise 6 section 4 chapter 1): Show that there is a $C^\infty$ map $f:D^3\to D^2$ with $0\in D^2$ as a regular value such that…
Adam Chalumeau
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Does there exist a compact submanifold of $\mathbb{R}^3$ whose fundamental group is $\mathbb{Z}^3$?

Does there exist a compact submanifold of $\mathbb{R}^3$ whose fundamental group is $\mathbb{Z}^3$ ? The question in the title is a generalization of the question that really interests me: Does there exist a connected finite set of unit cubes of…
Arshak Aivazian
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Fixed Points Set of an Isometry

I'm reading Kobayashi's "Transformation Groups In Riemannian Geometry". I'm trying to understand the proof of the following theorem: Theorem. Let $M$ be a Riemannian manifold and $K$ any set of isometries of $M$. Let $F$ be the set of points of $M$…
Sak
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Proving The Extension Lemma For Vector Fields On Submanifolds

I need some hints to prove the following lemma (John Lee's $\textit{Introduction to Smooth Manifolds 2nd Ed}$ p.201) : EXTENSION LEMMA FOR VECTOR FIELDS ON SUBMANIFOLDS: Suppose $M$ is a smooth manifold and $S\subseteq M$ is an embedded submanifold.…
Dubious
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Example of a submanifold $S\subseteq M$ that is an immersed submanifold is more than one way?

Known uniqueness results say that an embedded submanifold has a unique smooth structure making it an embedded submanifold with the subspace topology, and immersed submanifolds have a unique smooth structure making them immersed if we have a prior…
Strathbogie
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Searching for a bound for the integral of a 1-form along a loop.

Consider the submanifold $M$ of $\mathbb R^8$, with coordinates $(x_1,y_1,x_2,y_2,x_3,y_3,x_4,y_4)$, defined by the following equations $$x_1^2+y_1^2+x_2^2+y_2^2=1,$$ $$x_3^2+y_3^2+x_4^2+y_4^2=1,$$ $$x_1x_2+y_1y_2+x_3x_4+y_3y_4 = 0.$$ Consider a…
Hugo
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Are two homeomorphic hypersurfaces of the same smooth manifold also diffeomorphic?

Let $M$ be a smooth (connected, without boundary) manifold and $N_1$, $N_2$ be two smooth (connected, without boundary) hypersurfaces of $M$. Suppose $N_1$ and $N_2$ are homeomorphic. Can $N_1$ and $N_2$ be non-diffeomorphic? I am currently working…
Didier
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Smooth submanifold of $\mathbb R^6$. Not smooth submanifold of $\mathbb R^3$

So I'm pretty new to studying manifolds and have little to no background on differential geometry, but this is a question from lecture notes on a multivariable analysis unit: Show that $S:=\{(x^2,y^2,z^2,yz,xz,xy)|x,y,z \in \mathbb R,…
aaa
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How bad can the intersection of two totally geodesic submanifolds be?

Let $M$ be a complete Riemannian manifold and let $S_1,S_2 \subset M$ be totally geodesic submanifolds which are closed as subspaces of $M$. Question: Is $S_1 \cap S_2$ a submanifold of $M$? Or is it at least locally path connected? Notice that if…
abenthy
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$Sp(2n)$ is embedded in $GL(2n)$ and has dimension $2n^2+n$

Let $Sp(2n):=\{A\in\mathbb{R}^{2n\times 2n}\mid A^tA_0A=A_0\}$ be the group of symplectomorphisms from $(\mathbb{R}^{2n},\omega_0)$ to itself, where: \begin{gather} A_0 := \begin{bmatrix}{} 0 & I\\ -I & 0\\ \end{bmatrix}\in\mathbb{R}^{2n\times…
rmdmc89
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Intersection of two Lie subgroup is Lie subgroup ?

This question is already asked in here but i can't find a satisfactory answer. The question (in the title) arise from the following definition (i'm using Lee's smooth manifold p.156) : If $G$ is a Lie group and $S \subseteq G$, the…
Kelvin Lois
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Submanifold given by an immersion open onto its image

I was wondering if the following is true: Let $M,N$ be two manifolds such that $\dim M\leq \dim N$ and $f:M\rightarrow N$ an smooth immersion. Assume that for any open set $U\subset M$, $f(U)$ is open in $f(M)$, does it imply that $f(M)$ is a…
user109016
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The embedded submanifolds of a smooth manifold (without boundary) of codimension 0 are exactly the open submanifolds

I'm reading the proof of the following proposition in Lee's book Introduction to Smooth Manifolds: Proposition 5.1: Let $M$ be a smooth manifold. The embedded submanifolds of codimension $0$ in $M$ are exactly the open submanifolds. Lee proves…
Lazarus Frost
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Integral curves of a vector field not tangent to an embedded submanifold $S$

I got stuck on a problem. Let $M$ be a smooth $n-$dimensional manifold and let $S$ be a compact embedded submanifold. Suppose $V$ is nowhere tangent to $S.$ Prove that there exists $\epsilon>0$ such that the flow of $V$ restrict to a smooth…
Matteo Aldovardi
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Sections of a polar action are totally geodesic

Suppose $G\curvearrowright M$ is an isometric action of a Lie group on a complete Riemannian manifold $M$, and assume it is polar. This means that the action is proper and there exists a closed (hence complete) embedded submanifold $\Sigma\subseteq…
Johnny El Curvas
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