# 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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### 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 ...
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### 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!
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### 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 $k$-...
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### 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 $k$-...
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### 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 $p \in S$, ...
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### 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 ...
1 vote
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### 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 ...
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### 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 ...
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### 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) ...
1 vote
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### 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 ...
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### 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 ...
1 vote
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### 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 ...
1 vote
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### 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 ...
1 vote
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### 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 ...
1 vote
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### 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 ...
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### 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,...
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### 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 $\Sigma$ ...
1 vote
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### 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 ...
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### 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 ...
1 vote
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### 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
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### 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 ...
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### 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 ...
1 vote
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### 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 ...
1 vote
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### 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 ...
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### 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 ...
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### 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 ...
1 vote
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### 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)\...
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 ...
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 \...