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.
Questions tagged [submanifold]
745 questions
27
votes
2 answers
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
- 3,333
18
votes
1 answer
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
- 915
15
votes
1 answer
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
- 4,027
15
votes
1 answer
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
- 14,048
10
votes
1 answer
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
- 101
8
votes
3 answers
Professor Lee's Introduction to Smooth Manifolds Second Edition Lemma 10.35
I'm stuck trying to verify the proof given in the text. There are parts of the
hypothesis and proof that have nothing to do with where I'm stuck, so
in the interests of brevity, I'll give
only the part I'm having trouble with. We are given $M$, an…
Jeff Rubin
- 799
8
votes
0 answers
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
- 3,925
8
votes
2 answers
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
- 20,820
8
votes
1 answer
Show that Hopf foliation is a foliation.
Consider $S^3 := \{(z,w) \in \mathbb{C}^2:|z|^2 + |w|^2 = 1\}$ be the
unit $3$-sphere with equivalence relation
$$(z,w) \sim (z',w') \iff z' = e^{i \theta }z, w' = e^{i\theta} w$$
for some $\theta \in \mathbb{R}$.
My definition of…
user661541
8
votes
2 answers
Are initial submanifolds a thing?
Question
While studying Lie groups, I came across the notion of an "initial submanifold" mentioned in Definition 2.14. here. Seeing a technical definition like this (see details below) I consulted the great wisdom of the internet (looking at you,…
chickenNinja123
- 849
8
votes
1 answer
$Sp(2n)$ 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
- 10,709
- 3
- 35
- 94
7
votes
0 answers
What is the geometric explanation for why the shape operator is symmetric?
The shape operator at acts on the tangent space of a manifold at $p$:
$S_p: T_pM \rightarrow T_pM$
By:
$$S_p(v) = D_v(n(p))$$
That is, it sends the normal vector at $p$ to it's directional derivative in the direction of the input vector. The proof…
PhysicsIsHard
- 147
7
votes
2 answers
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
- 489
7
votes
1 answer
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
- 328
7
votes
0 answers
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
- 71