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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 ...
Joseph Kwong's user avatar
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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 ...
Fredrigo6's user avatar
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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 ...
Raul Fernandes Horta's user avatar
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Lee Smooth Manifolds Theorem 6.24 proof

I'm currently reading the book Smooth Manifolds by John M.Lee. There is something unclear to me in the proof of theorem $6.24$, the Tubular Neighborhood Theorem. The proof goes as follows. First, if $...
Mark's user avatar
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Eigenvalue decomposition of submatrix

for two rank- 1 matrices $\boldsymbol{A}$ and $\boldsymbol{A}_{\mathrm{sub}}$ I am interested in the relation between the eigenvalue decompositions (EVD) of ... \begin{align} &\boldsymbol{A} = \...
Dennis Marx's user avatar
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Fundamental equations of a Riemannian immersion

The Gauss, Codazzi-Mainardi and Ricci equations are the three fundamental equations of a Riemannian immersion. The Gauss equation inputs 4 tangent vectors, the Codazzi-Mainardi inputs 3 tangent ...
AlexInorbit's user avatar
1 vote
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When is a real analytic subvariety complex?

Let $X$ be a complex manifold and $Y\subset X$ a real analytic subvariety, not necessarily smooth. Suppose that there is a dense open subset $U\subset Y$ such that $U$ is a complex analytic subvariety ...
fgh's user avatar
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Normal to submanifold defined by a set of functions and divergence theorem

I have a submanifold $\Sigma(\mathbf{z})$ defined using a set of functions $\mathbf{q}(\mathbf{x}) = (q_1(\mathbf{x}), ..., q_m(\mathbf{x}))$. Let $\mathbf{z} = \mathbf{q}(\mathbf{x})$, and $\Sigma(\...
modsim's user avatar
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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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submanifold of constant curvature

If we have $N↪(M,h)$ wih $N$ totally umbilic submanifold and $M$ of constant curvature, I want to prove that $N$ is of constant curvature too. Attempt Using Gauss equation, we have: $R^M(X,Y,Z,W)=R^N(...
Jack's user avatar
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Numerically compute the volume of an implicit submanifold

Let $f:\mathbb R^n\to N$ be a smooth function and $M=f^{-1}(p)$ be the preimage of a regular value (more generally we could have the preimage of a submanifold transverse to $f$) and we can assume $M$ ...
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Justification of the induced orientation on a sub-manifold with boundary

In my study of orientation of sub-manifold, I tried to construct the induced orientation on the boundary of a sub-manifold and I would like to have your advice because in the book I have (Milnor’s ...
G2MWF's user avatar
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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 ...
PhysicsIsHard's user avatar
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Notion of distance for points lying outside of a manifold and in an ambient space

I am trying to understand how to extend the notion of distance for a given Riemannean manifold (M, g) lying in an ambient space X. I am thinking of the following $d_M$: $$d_M(x, y) = d(x, M) + d(x_{...
Cupitor's user avatar
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Proof that Torus is a 2-dim submanifold of $\mathbb{R}^3$ using parametrization

I want to show that the torus $T:=\{(x, y, z) \in\mathbb{R}^3 \mid (\sqrt{x^2+y^2}-R)^2+z^2=r^2\}\subset \mathbb{R}^3, 0<r<R<\infty$ is a 2-dimensional submanifold showing that $\forall p\in ...
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Does all unitary transformations of a matrix manifold form a manifold?

Let $\mathcal{M} \subset \mathbb{R}^{n \times r}$ be an embedded submanifold of $\mathbb{R}^{n\times r}$, is $\mathcal{MU}:= \{ MU|M \in \mathcal{M}, U\in \mathbb{R}^{r \times r}, U^{\top}U = I \}$ a ...
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Immersions are local diffeomorphisms

I am trying to understand the proof why the the image of an immersion and differentiable map between manifolds $f:M \to N$ is a submanifold. I am following the book of Guillemin, Differential Topology ...
GG314's user avatar
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Submanifold of matrix space

One can identify the space $M_{\mathbb{R}}(n,n)$ of real $n \times n$ matrices as $\mathbb{R}^{n^2}$. Consider the subset $S:=\{ A \in M_{\mathbb{R}}(n,n) : det(A)=1 \}$ and show it is a smooth ...
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Möbius strip is a smooth submanifold

I want to find a description of the Möbius strip without boundary as a submanifold. To be more specific, what I mean with "description". I know the following proposition: For a subset $M \...
Philip's user avatar
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Are Metric Balls on a Riemannian Manifold Embedded Submanifolds?

Let $(M,g)$ be a connected $n$-dimensional Riemannian manifold. We can define a topological metric on $M$ by $$ d_g(p,q) = \inf\{\text{Length}(\gamma) \mid \gamma \ \text{is a piecewise smooth curve ...
Algebro1000's user avatar
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Differential manifold connection with different notation

We aim to prove that the set {V(n, 2) = {(a, b) \in R^n \times R^n : |a|^2 = |b|^2 = 1 \text{ and } <a,b> = 0} is a smooth (2n-3)-submanifold of R^n \times R^n. This will be demonstrated by ...
Russell Hua's user avatar
1 vote
1 answer
41 views

Restriction of a compactly supported function on a bounded domain in a surface

Let $\Omega \subset \mathbb{R}^N$ be a bounded domain and let $P_k$ a plane of dimension $1\leq k<N$ in $\mathbb{R}^N$. Denote by $\sigma_k$ the surface measure in the surface $\Omega_k = \Omega\...
Lucas Linhares's user avatar
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Why are local models of blow-ups compatible by gluing with transition functions in the normal bundle?

I am trying to understand the blow-up along a submanifold as explained in Huybrechts "Complex Geometry An Introduction", p. 99, Example 2.5.2. For background, for $m \leq n$, we see $\mathbb ...
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3 votes
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How to determine a basis for the tangent space given a local trivialization.

I heard the following: For a smooth submanifold $M \subseteq \mathbb{R}^n$, given a local trivialization one can easily find a basis for the tangent space. I want to know how. So first I should ...
Peter's user avatar
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Understanding the Round Metric and various other Metric Tensors on $S^2$.

Suppose we are dealing with a $2$-dimensional sphere.For simplicity we consider $S^2\subset \mathbb R^3$.Now we can give a parametrization on the sphere via the map $\varphi:(0,2\pi)\times(0,\pi)\to \...
Kishalay Sarkar's user avatar
1 vote
1 answer
63 views

The image of a submanifold under a diffeomorphism is a submanifold

Let $M \subseteq \mathbb{R}^n$ be a smooth $k$-dimensional submanifold and let $A \subseteq \mathbb{R}^n$ be an open subset with $M \subseteq A$. Show that if $\psi$ is a diffeomorphism from $A \...
Peter's user avatar
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6 votes
1 answer
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A question on Lee's Proof of the Global Frobenius Theorem (Lemma 19.22)

I'm afraid this is a stupid question — I'm not a mathematician, so please correct me when I'll be saying something wrong — but I've been stuck at this point for so long that I thought it would be wise ...
atlantropa's user avatar
1 vote
0 answers
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Contradicting definitions for submanifold in Hirsch Differential Topology?

On page $30$ in Hirsch's Differential Topology one can find the sentence These are images of embeddings, and should be "submanifolds". This would imply that the definition of $C^r$-...
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Submanifold from commutative algebra perspective

It is well known that a Manifold is completely determined by its algebra of smooth functions. How can we describe a submanifold from an algebraic point of view?
Jack89's user avatar
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Cartesian product of two manifolds and tangent spaces

I have an $n$ dimensional parallelizable manifold $M$. I know that it is the cartesian product of two parallelizable manifolds $M_1$ and $M_2$ but I do not know these two manifolds, not even their ...
Doriano Brogioli's user avatar
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1 answer
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The image of a periodic nonconstant maximal integral curve is an immersed submanifold diffeomorphic to $S^1$

This is problem 9-1 from John Lee's Introduction to Smooth Manifolds. Suppose $M$ is a smooth manifold $X$ is a smooth vector field on $M$ and $\gamma$ is a maximal integral curve of $X$. Show that ...
nomadicmathematician's user avatar
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When the equivalent relaton $x\sim y \iff f(x)=f(y)$ gives a foliation?

Let $M$ be an $n$-dimensional smooth manifold, and let $f:M\rightarrow R^m$ be a smooth map with constant rank $m$ with $0<m<n$. Now consider the equivalent relation $\forall x,y\in M \ x\sim y \...
Rafael Rojas's user avatar
3 votes
0 answers
64 views

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 ...
Markus Klyver's user avatar
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0 answers
51 views

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-...
lolabol's user avatar
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3 votes
2 answers
162 views

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 ...
Liam's user avatar
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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 ...
Keshav Balwant Deoskar's user avatar
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1 answer
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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 ...
ccnptr's user avatar
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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 ...
Leonardo Moreira's user avatar
2 votes
0 answers
74 views

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 ...
Azur's user avatar
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2 votes
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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 $\partial U$...
P.Jo's user avatar
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Showing that a quadrifolium function from a circle is an immersion

I am trying to prove that a certain function is an immersion, but I am confused as to how to do it. I have a function $f: S^1 \rightarrow \mathbb{R}^2$ s.t. $(\cos\theta, \sin\theta) \mapsto (\sin2\...
fr_'s user avatar
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0 answers
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Global structure of submanifolds transverse to fibres of a fibre bundle

I try generalising the following "classical" result to the abstract fibre bundle (Rudolph-Schmidt, Differential Geometry and Mathematical Physics part I) I can state it as follows. Global ...
Parco Macelli's user avatar
3 votes
0 answers
45 views

Curvature calculation of $S^{2n+1}=U(n+1)/U(n)$ as a homogeneous space

I eventually want to check how the Hopf fibration $$ S^{2n+1}\to {\mathbb C}P^n $$ satisfies the Riemannian submersion formula, but I am frustrated that I could not even see how things work for $S^{2n+...
Three aggies's user avatar
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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 ...
DavideL's user avatar
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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_{...
H.F.A.'s user avatar
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0 answers
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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 ...
user167131's user avatar
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113 views

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 ...
user500357's user avatar
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53 views

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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