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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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### Homotopy relations between surfaces with boundary and submanifolds of dimension 2

Let $\Sigma(g,n)$ denote the oriented surface of genus $g$ with $n$ boundary components. Consider $X=\Sigma(g',n')$ a submanifold of $\Sigma=\Sigma(g,n)$, $g'\leq g$ and $n'\leq n$. Question: for the ...
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### Lee's Smooth Manifolds Problem 10-18

Lee's Introduction to Smooth Manifolds Problem 10-18 asks us to prove the following theorem: Theorem. Let $S$ be a properly embedded codimension-$k$ submanifold of $\mathbb R^n$. Then the following ...
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### Hodge star decomposition in non-diagonal manifold product

I'm studying differential forms and I came across the following problem. From what I learnt in another question, when a manifold can be decomposed as $X \times Y$, then the formula found there works ...
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### If a Banach manifold satisfies the Heine-Borel property, then does it have finite dimension? [closed]

Suppose $C$ is a topological Banach manifold, that is also a closed convex subset of a Banach space $E$, also, $C$ satisfies the Heine-Borel property: Every closed and bounded (with respect to the ...
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### Prove that the graph of $f(x,y)=\frac{\sin(x^2+y^2)}{x^2+y^2}$ is a smooth manifold

I would like to prove that $$A=\{(x,y,z) : z= \frac{\sin(x^2+y^2)}{x^2+y^2}, (x,y)\neq 0\}$$ is a smooth sub-manifold of dimension $2$. I think it is a direct application of the inverse function ...
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### Construct CMC hypersurface from CMC codimension $2$ submanifold

Say we have an isometric immersion $P^{n-2} \to \mathbb{R}^n$ with mean curvature vector $\vec{H}$. Assume $P$ has constant mean curvature, meaning $|\vec{H}|^2=\text{constant}$ on $P$. And let's say ...
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### Filling a Klein bottle with four-dimensional water

Can you fill a Klein bottle $K \subseteq \mathbb R^4$, embedded in four dimensional Euclidean space $\mathbb R^4$, with hyperraindroplets (hyperballs $B_\varepsilon$) from the outside? Will it enclose ...
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### Isometries of a submanifold with induced metric

I need help to find isometries of a submanifold of a semi-Riemannian manifold. To be crystal clear, let me start with what I mean by an isometry. $\textbf{Definition:}$ Let $(M,g)$ be a semi-...
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### Universal property of submanifold

Updated: Apologize. I realize that this question is not worth answering. It's known that the subspace of a topological space shares the universal property as follows: let $X, Y$ be topological spaces ...
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### Understanding the proof that the image of an embedding between manifolds is a submanifold + Alternate Proof

I'm trying to understand the proof of the following theorem on Page 17 of Guillemin and Pollack's Differential Topology: Theorem: An embedding $f : X \rightarrow Y$ maps $X$ diffeomorphically onto a ...
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### Adapted atlas of a submanifold

I feel like I didn't fully understand what an atlas of adapted charts really is. For example let's take $\mathbf{G}=SL(2,\mathbb{R})$ as a submanifold of $\mathbb{R}^4$ with the standard differential ...
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### Is the closed unit disk a regular surface?

A set $S \subset \mathbb{R^3}$ is a regular surface iff, for every point $p$ in $S$ there is a function $\phi: U \subset \mathbb{R}^2 \to W = V \cap S$, called a parametrization of $p$, where $U$ is ...
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### Local form of Lagrangians

Let $T^\star N$ be the cotangent bundle of some manifold $N$. I think it is a standard fact that, given a $1$-form $\mu$ on $N$, $\text{Graph}(\mu)$ is a Lagrangian submanifold of $T^\star N$. My ...
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### Definition of $C^k$ boundary in Evans book - Relations to regular domain?

I am trying to draw some relations between two (apparently) different definitions. The first one is from Evan's book: Let $U \subset \mathbb{R}^n$ be open and bounded. We say the boundary $\partial U$...
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### Divergence with respect to pull-back metric

Let $(M,G)$ be a Riemannian manifold of dimension $n$ and $S$ be a submanifold of $M$ of dimension $m$. Consider a vector field $X$ on $S$. $X$ is also a vector fieldd on $M$, so we can compute its ...
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### General solution to the eigenvalue problem of a $3×3$ symmetric Hessian defining curvature

The symmetric Hessian for an implicit surface defined from a field variable $c(x,y,z)$ in Cartesian space is,  \nabla^2 c = \begin{bmatrix} c_{xx} & c_{xy} & c_{xz} \newline c_{xy} & c_{...
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### Proof that inclusion map is smooth when submanifold is defined by slice charts

I would like to ask if my thought process in answering the question in the Title is correct. I know there are many posts on this general topic: e.g. Is the inclusion map always smooth?, When is an ...
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### Existence of closest point projection to embedded submanifold

I am currently trying to understand closest point projections to a given embedded submanifold $M$ of $\mathbb{R}^n$. In Lee's "Smooth manifolds" I read about the existence of tubular ...
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### Can the Poincare half plane be isometrically embedded into $\mathbb{R}^3$ (with Euclidean metric)?

Can the Poincaré half plane be isometrically embedded into $\mathbb{R}^3$ (with Euclidean metric)? I am self-studying Riemannian geometry and have found it harder to imagine a surface with negative ...
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