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.
643
questions
0
votes
0
answers
40
views
How does the Riemannian Hessian change when changing the metric?
Let $M$ be a smooth Riemannian manifold with metric $\langle\cdot,\cdot\rangle_x$, and consider a smooth function $f:M\to\mathbb{R}$. Suppose that we consider another Riemannian metric $\langle\cdot,\...
- 315
0
votes
0
answers
21
views
Trouble with proving certain equality of the covariant derivative of a pseudo-riemannian manifold
I am trying to fill the details from this paper (it's free available) and I get stuck proving the equation (3.16) from the mentioned article. Let me give some context.
Part 1. Setting the notation.
...
- 869
2
votes
0
answers
49
views
The Operator Algebra of Hypersurface Deformation Operators
Let $\Sigma$ and ${M}$ be smooth manifolds with $\mathrm{dim}(\Sigma)<\mathrm{dim}({M})$, as well as the coordinate charts $\xi$ on $\Sigma$ and ${x}$ on ${M}$. Let further be $\Phi\in{C}^{\infty}(\...
- 76
1
vote
1
answer
58
views
The cuspidal cubic $\{(t^2,t^3)| t\in\mathbb{R}\}$ is not a regular submanifold of $\mathbb{R}^2$.
I want to figure out why $\{(t^2,t^3)| t\in\mathbb{R}\}$ is not a regular submanifold of $\mathbb{R}^2$. There are some posts discussing this (like this post). But I do not quite understand them. So I ...
- 837
0
votes
0
answers
23
views
Bounds on radius given positive second fundamental form
Given a closed hypersurface $\Sigma^{n-1}$ in $\mathbb{R}^n$. If the second fundamental form $II\geq 1$, then does $\Sigma$ lies within a ball of radius 1?
Any references will be appreciated. Thanks!
- 91
0
votes
0
answers
93
views
Equivalence of definitions of a submanifold
I have a question adressing the "abstract" definition of a submanifold and the version in the special case of $\mathbb{R}^n$. This is what I mean:
I looked up these two definitions of a $k$-...
- 41
0
votes
0
answers
50
views
Equivalence of definitions of a submanifold [duplicate]
I have a question adressing the "abstract" definition of a submanifold and the version in the special case of $\mathbb{R}^n$. This is what I mean:
I looked up these two definitions of a $k$-...
- 41
3
votes
1
answer
100
views
Upper bound on second fundamental form
Consider a manifold $\mathcal{M}$ embedded in $\mathbb{R}^d$ and let $V$ and $W$ be two vector fields on $\mathcal{M}$.
As usual, we define the second fundamental form to be $ \mathrm{I\!I}(V, W) = (\...
- 411
1
vote
0
answers
37
views
Proof on invariant submanifold optimal control problem
I need help to understand a proof about invariant submanifolds. I wouldn't ask but I have been trying for a week unsuccessfully and I am truly getting crazy over it.
The system is the classic state-...
- 11
1
vote
0
answers
66
views
Figure-8 is not an embedded submanifold of $\mathbb{R}^2$
Define the figure-8 curve $S$ to be the image of the smooth map $\beta : (-\pi,\pi) → \mathbb{R}^2$ given by $\beta(t)=(\sin 2t, \sin t)$.
I am trying to prove that $S$ is not an embedded (or regular) ...
1
vote
1
answer
88
views
¿Pullback commutes with lie derivative?
Everything is smooth. Let $(M,g)$ be a riemannian manifold and $S\subseteq M$ a submanifold, $X$ be a vector field of $M$ such that $X|_S$ is smooth and $X_p\in T_pS$ for every $p\in S$, $\mathcal{L}$ ...
- 341
0
votes
1
answer
56
views
When a set $F^{-1}(0)$ isn't a smooth submanifold [closed]
Let F be a smooth function: $\mathbb{R}^n \to \mathbb{R}$. If the rank of a function is not constant, does it imply that $F^{-1}(0)$ isn't a smooth submanifold. And if it doesn't, is there some way to ...
- 11
0
votes
1
answer
47
views
How to tell if a set is a submanifold
I have been reading a paper lately. It said that, "consider the subset $S=\{R\in SO(3):2<tr[R]\le3\}$. Since $−1 ≤ tr[R] ≤ 3$ on $SO(3)$, we express the set $S=\{R\in SO(3):2<tr[R]\le3\}$= $...
0
votes
1
answer
45
views
Last step in the proof of $ G(F):=\{(p, F(p); p \in M\}$ is an embedded submanifold of $M \times N$ of dimension $m$, $F:M\rightarrow N$ smooth
Consider the definition of embedded submanifold as follows :Let $M$ be a smooth manifold of dimension $m$ and $S\subseteq M$. $S$ is an embedded manofold of dimension $k$ if for every $p \in S$, ...
- 469
0
votes
0
answers
24
views
Is the projection of an embedded submanifold of Euclidean n-space onto a subset of its coordinates a manifold?
Suppose I have a smooth (overlap maps are $C^\infty$) embedded submanifold $S \subset R^n$ of dimension $d > 2$. Let $\pi : S \rightarrow R^2$ be the projection of $S$ onto its first two ...
- 577
1
vote
0
answers
56
views
How to prove subbundle example (10.33(c) in Lee’s Intro to Smooth Manifolds) in boundary case
Prove that $TS$ is subbundle of $TM|_S$ for immersed submanifold
asked how to show that $TS$ is a subbundle of $TM|_S$ when $S\subseteq M$ is an immersed submanifold. However, the origin of this ...
- 727
0
votes
0
answers
45
views
Boundary of the boundary of an oriented compact manifold-with-corners
Suppose I have an oriented, compact manifold-with-corners $M$ (see Jack Lee’s Introduction to Smooth Manifolds, pages 417-419). It is not hard to see (using the results of these pages) that its ...
- 3,306
0
votes
0
answers
27
views
Metric that makes distribution orthogonal to each other
Let $M$ be a manifold and $Q\subseteq M$ a submanifold. Suppose that there is an integrable distribution (sub-bundle) $E$ of $TM|_Q$ such that $TM|_Q=TQ\bigoplus E$. Can we guarantee a (Riemannian) ...
- 589
1
vote
1
answer
47
views
Why is the connection of a submanifold equal to the connection on the ambient space?
I have a question about the proof of the following statement.
Let $M$ be an oriented hypersurface with constant mean curvature in $\mathbb{R}^{n+1}$ and with second fundamental form $B$. Let $v$ be ...
- 395
0
votes
1
answer
57
views
Restriction of Differential Form On a Connected and Compact Submanifold
Show that, conversely, if $M \subset \mathbb{R}^3$ is a compact and connected submanifold with the proper
$$
\left.(x d x+y d y+z d z)\right|_M=0,
$$
then $M$ is one of the spheres centered at the ...
- 121
1
vote
2
answers
140
views
What does it mean to induce a Riemannian metric on an evolving hypersurface immersed in a Riemannian manifold?
Oftentimes in some journal articles, I encounter a statement like "Let $F:M^n\times[0,T]\to N^{n+1}$ be a one-parameter family of immersions in a Riemannian manifold $(N,g)$ and let $g_t$ be the ...
- 215
1
vote
0
answers
26
views
The relationship of tangent space between submanifold and manifold.
I was confused when I prove the next Proposition appeared in the book Introduction to smooth manifolds by Lee.
Suppose $M$ is a smooth manifold with or without boundary ,$S\subset M$ is an immersed ...
- 128
1
vote
1
answer
35
views
Manifolds - Inverse Function Theorem Form?
For context, I am reading Appendix F is Milnor's book on Sullivan's No Wandering Theorem and this is where he uses the following manifolds result:
Let $F:X \rightarrow Y$ be a smooth function between ...
- 125
1
vote
0
answers
44
views
When is the "flat limit" of a submanifold not a submanifold?
We have a compactness result, where the "flat limit" of an integral current is itself an integral current. (with some conditions)
Now I am curious about submanifolds. I am expecting that an ...
- 23
2
votes
1
answer
240
views
Riemannian volume forms on a family of surfaces evolving by IMCF
Fix a closed hypersurface $\Sigma$ in a Riemannian $3$-manifold $(M,g)$. Let $\Sigma_t$ be a family of closed hypersurfaces evolving from $\Sigma$ by inverse mean curvature flow (IMCF) in $M$. That is,...
- 215
2
votes
2
answers
63
views
Integration over a closed surface in a Riemannian $3$-manifold
Let $\Sigma$ be a closed surface in a Riemannian $3$-manifold $(M,g)$. I was thinking about the validity of writing
$$\int_\Sigma H^2 d\mu_\Sigma,\tag{1}$$
where $H$ is the mean curvature of $\Sigma$ ...
- 215
1
vote
0
answers
41
views
What is the difference between a boundary *existing* (or rather, *not* existing) and a boundary *being empty*?
I'm confused about the concept of a "boundary" in topology – specifically, in studying manifolds. What is the difference between a boundary existing (or rather, not existing) and a boundary ...
- 4,580
5
votes
2
answers
162
views
The boundary of any manifold that is itself a boundary of a higher-dimensional manifold must always be empty?
I have seen it asserted several times that it is well-known that the boundary of any manifold that is itself a boundary of a higher-dimensional manifold must always be empty. Yet, despite it ...
- 4,580
1
vote
0
answers
55
views
Diagonal is submanifold, Local zero set description of submanifolds.
Let $M$ be a smooth manifold and $\Delta_M=\{(p,p)\,\vert\, p\in M\}$ be the diagonal of $M$. Then $\Delta_M$ is a submanifold of $M\times M$.
I have proved this by showing that $\mathscr{i}:M\to M\...
- 413
7
votes
1
answer
324
views
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 ...
- 727
2
votes
1
answer
86
views
On the definition of a regular submanifold
A subset $S$ of a manifold $N$ of dimension $n$ is a regular submanifold of dimension $k$ if for every $p \in S$ there is a coordinate neighbourhood $(U,\phi)$ of $p$ in the maximal atlas of $N$ such ...
- 245
0
votes
0
answers
41
views
When does a map become an immersion? When does a subspace of matrices space become a submanifold?
We identify the set $M_4(\Bbb R)$ of all real square matrices of order 4 with the Euclidean space $\Bbb R^{16}$. Let $M$ be the subspace of $M_4(\Bbb R)$ which consists of all matrices $A=(a_{ij})_{i,...
- 31
0
votes
1
answer
36
views
Adapted expression to a submanifold for a smooth function
I am trying to prove the following:
Lemma. Let $S$ be an embedded smooth submanifold of a smooth manifold $M$, let $f\in C^\infty(M)$ be such that $f|_S=0$, and pick $p\in S$. There exists adapted ...
1
vote
0
answers
78
views
Misunderstanding of a theorem, that seems to state that the image of a smooth map of constant rank with compact domain is an embedded submanifold
Context:
I'm studying smooth manifolds, and I'm trying to better understand what can be said about the images of smooth maps $\varphi: M \to N$ of constant rank $r$.
I know that from the constant rank ...
- 11
0
votes
0
answers
37
views
Proving embedded submanifolds have submanifold charts
I am trying to prove the following statement:
A subset $K$ of an $m$-dimensional $M$ is an embedded submanifold of dimension $k$ if and only if around each $p\in K$, there exists a chart $(U,\phi)$ of ...
- 209
1
vote
0
answers
46
views
Intersection number of submanifolds with boundary
Let $N$ be an $n$-dimensional compact oriented manifold without boundary, and let $X$, $Y$ be compact oriented submanifolds of complementary dimensions $k$ and $n - k$, with boundary but such that $X \...
- 23
3
votes
1
answer
118
views
Guillemin & Pollack Exercise 1.8.14 (Generalized inverse function theorem)
I have a question regarding Exercise 1.8.14 in Guillemin & Pollack. (I think this question also applies to the answers here.) Here's the exercise:
Inverse Function Theorem Revisited. Use a ...
- 3,253
0
votes
0
answers
28
views
Restricting open neighborhood of submanifold
let $M$ be a smooth manifold, $N$ be a closed embedded submanifold of $M$ with $dim N< dim M$ and $U$ be an open neighborhood of $N$ in $M$. I wonder when it´s possible to find a smaller open ...
- 23
1
vote
0
answers
55
views
Understanding the $k$-dimensional totally-geodesic submanifolds in the Poincare Ball model of the Real Hyperbolic space
I was looking at the submanifolds in the Poincare Ball model of the real hyperbolic space. It has been mentioned in various places that the totally geodesic hypersurfaces are the parts of the sphere ...
- 3,987
0
votes
0
answers
18
views
Show that $M = \{(r \cos(\sigma), r \sin(\sigma), \sigma ): r \in (0,1), \sigma \in \mathbb{R}\}$is a submanifold of $\mathbb{R}^3$ (Helicoid)
My tutor mentioned that i should use the regular value theorem for this. I kind of know how to do this for sets like $M=\{x \in \mathbb{R}^3 : g(x)= 0 \}$. However i dont know how to use it on this ...
1
vote
1
answer
90
views
Clarification on the Definition of a Submanifold of $\mathbb{R}^N$ (and the differentiability of it's charts)
Review of other questions on this site
In the following, my question is not asked/answered, with the last linked question being perhaps the most relevant
Clarification about Definition of Immersed ...
- 470
0
votes
0
answers
40
views
Is any minimal hypersurface also a minimal submanifold?
Recently I learned that a minimal submanifold is one whose mean curvature vector $\vec{H}$ vanishes identically, which makes me wonder whether a minimal hypersurface qualifies as a minimal submanifold ...
- 215
2
votes
1
answer
95
views
Explicit Lie group embedding of $O(n)$ into $SO(n+1)$
I am working on an exercise that asks to find an explicit group embedding of $O(n)$ into $SO(n+1)$. The answer to another question (Embedding between Lie Groups) suggests that the map can be given by
$...
- 4,721
0
votes
0
answers
45
views
Other ways to specify a submanifold?
Suppose $M$ is a smooth manifold. I am intersted in the different types of data I can use to specify a smooth submanifold $N\subseteq M$.
For example, if $f:M\to \mathbb{R}^n$ is a smooth map, then ...
- 379
1
vote
0
answers
86
views
Projection on compact submanifold embedded in Euclidean space.
For a compact submanifold $\mathcal{M}$ embedded in Euclidean space, its sectional curvature is positive, given $x,y\in \mathcal{M}, \eta \in T_x\mathcal{M}$, where $T_x\mathcal{M}$ denote the tangent ...
- 180
4
votes
1
answer
83
views
compact manifold
I am study for Ph.D. qualification examination in geometry, i don't know solve this question it's from Utah university 1999 test:$$ \\$$
"Let $M$ be a surface compact without boundary embedded ...
- 53
3
votes
1
answer
85
views
The image of a Riemannian submanifold under a diffeomorphism
Let $\Sigma$ be a compact submanifold of a Riemannian manifold $(M,g)$, let $h$ be the induced metric on $\Sigma$, and let $\Phi:M\to M$ be a diffeomorphism. $h_\Phi$ will denote the induced metric on ...
- 215
1
vote
0
answers
44
views
How to compute covariant derivative in immersed submanifold?
This is a part of an exercise in Riemannian Geometry by do Carmo in page 140.
Let \begin{align*}f:\mathbb{R}^2\to \mathbb{R}^4,(\theta,\varphi)\to &~(\cos\theta,\sin\theta,\cos\varphi,\sin\varphi)\...
- 363
3
votes
1
answer
93
views
Why must integral manifolds be immersed submanifolds?
Given a manifold $M$ and a distribution $D \subset TM$, an integral manifold of $D$ is defined as a nonempty immersed submanifold $N \subset M$ such that $T_pN = D_p$ for all $p \in N$.
Why are we not ...
- 4,721
3
votes
0
answers
49
views
Simpler argument that embedded submanifold has unique smooth structure
I am trying to prove the following statement:
Let $f:X\to Y$ be a embedding of differentiable manifolds. Then there is a unique manifold structure on $S:=f(X)$ such that the inclusion map $S \...
- 501