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.
Questions tagged [submanifold]
670 questions
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How do I find a,b such that M_a,b is a submanifold?
I have the following problem:
Let $$M_{a,b}=\{(x,y,z): x^1+y^2=z^2,\,\, z=ax+b\}$$ Investigate for which (a,b) this is a submanifold.
I thought that we could look at the following function $$F(x,y,z)=(x^2+y^2-z^2,\,\,\,z-ax-b)$$ where…
user123234
- 2,885
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What is the relation between totally real submanifold and Lagrangian submanifold?
By definition, for a complex manifold M, totally real submanifold X of M is satisfying
1) $2 dim X$ = $dim M$ and
2) $T_pM \cap J T_pM =\{0\}$ for $\forall p \in X $with integrable complex structure $J$.
I saw the question Lagrangian submanifold…
twinkling star
- 301
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How can I show that the following set is not a submanifold?
I have the following problem:
Show if the set $M_{a,0}=\{(x,y,z):x^2+y^2=z^2, z=ax\}$ is a submanifold for $11$.
Geometrically I see that if $1
user123234
- 2,885
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Can the Lebesgue integral $\int _U f(x)\,dx$ in $\mathbb {R}^d$ exist if $\int _W f|_W \,dS$ doesn't exist?
Let $M \subset \mathbb {R}^d$ be a $n$-dimensional submanifold, $W$ is a coordinate chart of $M$ with the map $\varphi : U \to V$ and $f \colon U \to \mathbb {R}$ is a continuous function function.
Can the Lebesgue integral $\int _U f(x)\,dx$ in…
Philip730
- 395
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How to tell if a set is a submanifold
I have been reading a paper lately. It said that, "consider the subset $S=\{R\in SO(3):2
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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 coordinates and assume that the rank of $\pi$ (i.e. the rank…
user167131
- 587
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Why is it a submanifold or not?
I am looking at the following set $$M:=\{(x,y) \in \Bbb R^2:x^3 =y^3\}$$
I know the equivalent definition but I can't explain why this should be a submanifold or not. Can please someone gives me some basic explanation, thank you!
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Short question about submanifolds
Which of these subsets $M \subset \mathbb{R}^2$ is a submanifold?
a. $M = \{(x,y)\,|\, x^2+y^2=1\}$
b. $M = \{(x,y)\,|\, x=0 \mbox { oder } y= 0\}$
c. $M = \mathbb {Q}^2$
I'm kind of sure that a is a submanifold and that c is not. I'm not sure about…
Philip730
- 395
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Synge inequality
Let $f : M \rightarrow N$ be an isometric immersion, and let $\gamma :[0,1] \rightarrow M$ be a smooth curve such that $f\circ\gamma $ is a geodesic in $N$ . Show that $ \gamma$ is a geodesic in M and, for each plane $ \sigma \subset T_{\gamma…
Kevin
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