# 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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### Bound for the genus of the incompressible surfaces that can be embedded in a closed compact orientable irreducible 3-manifold

Let $M$ be a closed compact orientable irreducible 3-manifold. I would like to know if there exists a bound $n\in \mathbb{N}$ such that every incompressible surface embedded in $M$ has genus lesser ...
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### What is the Jacobian of det???

In some questions of submanifold, I want to use determinant as a $C^{\infty}$Map, and show that, for example, $det^{-1}\{-1\}$ is a submanifold of $M_{2n}(\mathbb{R})$(this may be not true). However, ...
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### Second Fundamental Form of the Graph of a Function of Higher Codimension

Let $f:\mathbb{R}^n\to\mathbb{R}^m$, $m\geq 2$, be a smooth function. I would like to find an explicit description for the second fundamental form of the graph of $f$ in terms of the Hessians and ...
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### Submanifolds with codimension $0$

So pretty much as the title said, I'm supposed to find all submanifolds of $\mathbb{R}^n$ with codimension $0$. I haven't got too far but here are some of my thoughts: I started with intuitive ...
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### Why is this map no an immersion at 0

I wonder why the map $f:\mathbb R\to \mathbb R^2$ defined by $f(t)=(t^2, t^3)$ is not an immersion at 0. Isn't the derivative of $f$ is $(2t, 3t^2)$ which is injective? Thank you
1 vote
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### Why the definition of smooth boundary defined like this and how does it imply the interior ball property?

I am trying to understand what the definition of smooth boundary is. From the following lectures notes on analysis 3 : So intuitively, a smooth boundary ( in this case , it is a $C^1$ boundary) ...
1 vote
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### Zeros of second fundamental form

Let $M$ be a submanifold of $N$ and suppose that $M$ and $N$ are equipped with arbitrary connections $\nabla^M$ and $\nabla^N.$ Let $A = \{ X \in TM : \mathrm{I\!I}(X,X)=0 \},$ where $\mathrm{I\!I}$ ...
1 vote
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### Surface with a flat umbilic point that has points on each side of the tangent plane to it

The problem is the following: Find a surface S that has a flat umbilical point P (this means $K = H = 0$ at P) such that for every open set $U \in \mathbb{R}^3, P\in U$, there are points of $S \cap U$...
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### smooth maps between submanifolds, the image of tangent space under differential is contained in a tangent space

I want to show : This is from "Mathematical analysis" by Andrew Browder. This is not a manifolds text so we have only defined submanifolds on $R^n$ using the local immersion definition. I ...
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### Higher order expansion of hypersurface about a point (beyond second fundamental form/extrinsic curvature)

Consider a smooth, compact $(d-1)$-dimensional hypersurface $S$ without boundary embedded in $\mathbb{R}^d$. The surface $S$ can be described as the graph of a function $f(x_1,x_2,\cdots,x_{d-1})$. ...
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### What is the "derivative" of a family of submanifolds?

Suppose that $N(t)$ is a family of k-dimensional submanifolds in an n-dimensional manifold $M$, with $n > k$. Then $N(t)$ is a path in the space $\mathcal{M}$ of all k-dimensional submanifolds. I ...
1 vote
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### Confusion regarding the construction of a parameterization for a manifold

I am trying to prove that the differential of the inclusion map $i: X \rightarrow Y$ of a submanifold $X \subset Y$ $$di_x: T_x(X) \rightarrow T_x(Y)$$ is the inclusion map. This question has been ...
1 vote
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### Can we know that a map is a submersion with using only its level sets

Let $M$ be a manifold of dimension $2n$, $f:M\rightarrow\mathbb{R}^n$ a smooth map. We know that if $f$ is a submersion, then for any $c\in f(M)$, $f^{-1}(c)$ is a $n$-submanifold of $M$. But what ...
1 vote
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### Lee Smooth Manifolds, why does the Whitney Approximation Theorem fail when the co-domain has non-empty boundary?

I am trying to study chapter 6 of Lee's Introduction to Smooth Manifolds. In a remark after the Whitney Approximation Theorem, Lee stated that this theorem do not hold because it might not be possible ...
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### Characterizing accelerations of paths in a submanifold

$\newcommand{\R}{\mathbb{R}}$ $\newcommand{\M}{\mathcal{M}}$ $\newcommand{\Hs}{\operatorname{Hess}}$ $\newcommand{\al}{\alpha}$ This is a curious inquiry: Let $f:\R^N \to \mathbb{R}^n$ be a smooth map,...
1 vote
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### Separating two non-transversal manifolds via deformation

I was reading Guillemin and Pollack's book to self-study differential topology, where I encountered the following two related problems (Exercise 5 and 6 in Section 2.3): (Exercise 5) Assume that one ...
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