# 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.

102 questions
2answers
45 views

### Question about connected manifold

I need some tip to prove the following: If $N^{n}$ is a connected manifold and $M^{m}$ is a closed submanifold of $N$, such that $n-m\geq 2$, then $N-M$ is connected. I am supposed to use ...
1answer
13 views

### Does there exist an open neighborhood $U$ of $S$ and a smooth map $U\to S$ which is a retraction?

Suppose $M$ is a smooth manifold and $S$ a smooth embedded submanifold of $M$. Does there exist an open neighborhood $U$ of $S$ and a smooth map $U\to S$ which is a retraction?
0answers
17 views

### Intersection of two dimensional submanifolds of $\mathbb{R}^3$ with disjoint normal spaces is a one dimensional submanifold

Let $M_1, M_2$ be two two dimensional submanifolds of $\mathbb{R}^3$ such that the normal spaces $N_p$ $$N_p(M_1) \cap N_p(M_2) = \{0\}$$ for every point $p \in M_1 \cap M_2$. Then $M_1 \cap M_2$ is a ...
0answers
27 views

1answer
33 views

### Does a basis for a Lie algebra of a Lie group $G$ depend on whether $G$ is embedded?

My professor recently gave the problem to find a basis for a lie algebra of a given embedded lie subgroup. The problem stressed that the lie subgroup was embedded (which was clear from the definition ...
1answer
90 views

### Smooth images of manifolds are immersed?

in various papers in symplectic geometry, I have encountered the following argument. Statement: Suppose $f: M \rightarrow N$ is a smooth map of constant rank. Then its image $f(M)$ can be equipped ...
1answer
25 views

### Are the matrices of nullity at most one and non-negative determinant a submanifold with boundary?

Let $\text{GL}_n^+$ be the group of real invertible $n \times n$ matrices with positive determinant. Set $S=\text{GL}_n^+ \cup \{A \, | \, \text{rank} (A) = n-1 \}$. Does $S$ form a a submanifold ...
0answers
28 views

0answers
15 views

### For $\Phi(x, y)=x^{2}-y^{2}$ can $\Phi^{-1}(0)$ be given a topology and a smooth str. s.t it is an immersed sub manifold?

In the book of Lee, introduction to smooth manifolds, at page 123, it is asked that \begin{array}{l}{\text { Let } \Phi : \mathbb{R}^{2} \rightarrow \mathbb{R} \text { be defined by } \Phi(x, y)=x^{...
0answers
31 views

### For which values of $a$, is ${M_{a}=\left\{(x, y) : y^{2}=x(x-1)(x-a)\right\}}$ a sub manifold?

In the book of Lee, introduction to smooth manifolds, in page 123, it is asked that \begin{array}{c}{\text { For each } a \in \mathbb{R}, \text { let } M_{a} \text { be the subset of } \mathbb{R}^{2} ...
1answer
22 views

1answer
23 views

### What does continuity of the determinant say about the value of a submatrix on neighborhoods of $M_{m\times n}(\mathbb{R})$

I am following Lee's book on smooth manifolds. On pages 19 and 20 he writes the following: Suppose $m < n$, and let $D_m \subset M_{m\times n}(\mathbb{R})$ be the set of real $m\times n$ ...
0answers
80 views

### Ricci equation is trivial in codimension $1$

Let $f:M\to\overline{M}$ an isometric immersion and assume $\dim(M)=\dim(\overline{M})-1$. I'm asked to show that the Ricci equation offers no information. I guess what I have to show is that the ...
1answer
68 views

### The square is not a submanifold of $\mathbb{R}^2$

Leb $X$ be the square in $\mathbb{R}^2$ $X = \{(x,y) \in \mathbb{R}^2 : |x| + |y| = 1\}$ It's so easy to show that $X$ is a differentiable manifold of dimension one. But, it's not possible that $X$...
0answers
34 views

### Tangent spaces of two transverse subspaces are transverse subspaces

I am very new to differential geometry and was thrown this very long question: Suppose that two subspaces $V$ and $W$ of $\mathbb{R}^n$ are transverse (so $\text{Span}(V,W)=\mathbb{R}^n$). Let $O$ be ...
0answers
22 views

### Show that all solutions are of the form $(t,t^2,t^3)$

I want to show that there exists a global map of a submanifold described as the solution of the equations below. So I found one map, but I need to know if it contains all possible solutions. How does ...
0answers
52 views

0answers
13 views

### coordinate systems produce submanifolds

In his book "Semiriemannian geometry with applications to relativity", Barrett O'Neil says on page 16 under definition 26 that "coordinate systems produce submanifolds.If T:U->R^n is a coordinate ...
0answers
45 views
+100

### Anti-de Sitter space: Is the universal cover of $\text{AdS}_n$ a submanifold of $\text{AdS}_{n+1}$?

$$\text{AdS}_n = \{\vec x\in\mathbb R^{n-1,2}\mid\vec x\cdot\vec x=-1\}$$ Using an orthonormal basis $\{e_i\}$ for $\mathbb R^{n-1,2}$, with ${e_1}^2={e_2}^2=-1$ and $(e_{\geq3})^2=1$, we can ...
1answer
68 views

2answers
80 views

### Why $\mathbb R$ with the chard $t\longmapsto t^3$ is not a submanifold of $\mathbb R^2$ ?

I'm not very used to Manifold. I'm a little bit confuse with something. 1) Let $\phi:t\longmapsto t^3$ with $t\in \mathbb R$. So $(\mathbb R,\phi)$ is a smooth manifold. Now, why this is not a ...
0answers
39 views

### Regular level set and submanifold

I read a textbook and it says: A level set $S = \{x:F(x)=c\}$, $F: \mathbb{R}^n\rightarrow\mathbb{R}^k$ being analytic. $S$ is regular if it is nonempty and the Jacobian matrix of $F$ has maximal ...
1answer
52 views

### Why the 'gradient of the diffeomorphism at a point in the surface' perpendicular to the surface at that point?

This question is related to these two questions of mine: Intuition or motivation for the definition of an hypersurface. What are we actually trying to define? and Understanding this very generic ...
0answers
29 views

1answer
71 views

### Differentiability of functions on Manifolds

Let $f:\mathbb{R}^n\to\mathbb{R}^m$ be smooth, and suppose $M$ and $N$ are submanifolds of $\mathbb{R}^n$ and $\mathbb{R}^m$. Moreover, assume that $f(M) \subseteq N$. I want to prove that $f:M\to N$ ...
1answer
52 views