# Tagged Questions

For questions on manifolds of dimension $n$, a topological space that near each point resembles $n$-dimensional Euclidean space.

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### If $M^n$ is a submanifold of $\mathbb{R}^{n+k}$ such there are L.I. vector fields $v_1,\ldots,v_k$ such that $v_i \perp T_pM$ then $M$ is orientable

If $M^n$ is a submanifold of $\mathbb{R}^{n+k}$ such there are L.I. vector fields $v_1,\ldots,v_k$ such that $v_i \perp T_pM$ for every $p \in M$ then $M$ is orientable. My attempt is: Once $M^n$ ...
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### Viewing complex projective space as a Grassmannian manifold

Complex projective space $\mathbb{CP}^n$ carries the structure of a complex manifold of dimension $n$, hence has the underlying structure of a real manifold of dimension $2n$. It is the set of complex ...
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### Distributions on submanifolds

I am beginner in differential geometry. I stuck with the concept of distributions(like invariant, anti invariant, slant) on submanifolds. Can you explain what are distributions on submanifolds? If ...
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### Poincaré–Bendixson theorem on the torus

I was reading the paper A Generalization of a Poincaré-Bendixson Theorem to Closed Two-Dimensional Manifolds by Arthur J. Schwartz which proves the following theorem: THEOREM. Let $M$ be a ...
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### Definition of manifold which are subset of euclidean space

According to Guillemin and Pollack "X(which is a subset of R^n)is a k-dimensional manifold if it is locally diffeomorphic to Rk , meaning that each point x possesses a neighborhood V in X which ...
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### sectional curvature of hyperbolic space

For a pair of $(X,Y)$ of linearly independent vectors in $T_pM$, $p\in M$, the sectional curvature is defined as $$K_p(X,Y)= - \frac{<R(X,Y)X,Y>}{|X|^2 |Y|^2 - <X,Y>^2}$$ The problem I'm ...
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### addition of two differential forms with different degrees

Does it make sense to add two differential forms with different degrees like $dx+dx\bigwedge dy$? If yes, what's the arguments of it? I ask this because in text book, the vector space, $\Omega^*(M)$, ...
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### Construct a diffeomorphism $\psi: B_1 \to \epsilon\text{-neighborhood of } K$, where $K$ is a subset of a smooth manifold.

I'm currently working through a paper by Brezis on the topology of Sobolev spaces. Right now I am having trouble understanding the following note made by Brezis. Let $M$ be a compact and smooth ...
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### Transformation of vector from vector field coordinates

Consider an n-dimensional manifold with coordinates $x_1, x_2, \dots, x_n.$ Suppose we have a vector field defined on this manifold $V : \bar v = \bar v(\bar x).$ Let us perform a homogeneous ...
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### How to got there are only two kinds of 1-dim manifold without boundary

I just know a conclusion that all 1-dim manifolds without boundary is homomorphism to $S^1$ or $\mathbb{R}$ , but I don't know how to prove it . Why is so ?
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### Do I have the right idea about affine connections?

On a smooth manifold $M$, a vector field is a smooth map $X : M \to TM$, where $TM$ is the tangent bundle of $M$. If $\chi(M)$ denotes the space of vector fields on $M$, an affine connection $\nabla$ ...
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### Vector fields (on a manifold) and terminology

I read in several books (Do Carmo, Riemannian Geometry or John M. Lee, Smooth manifolds) that a vector field $X$ on a smooth manifold $M$ is a mapping which associates to each point $p \in M$ a ...
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### A exercise of Riemannian geometry . [closed]

In picture below,I don't know how to start the second question . It is obvious that the isometry of $R^3$ keep the dimension , so there exist such isometry. But seemly, it is too simple . Besides, ...
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### How to prove that two compact 1-dim Riemannian manifolds with same length must be isometric?

How to prove that two compact 1-dim Riemannian manifolds with same length must be isometric ? I know the compact 1-dim manifold must be homeomorphism to $S^1$ , but how to do a specific isometric ?
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### Where does $f(r,\varphi,\theta)=(r\sin{\theta}\cos{\varphi},r\sin{\theta}\sin{\varphi},r\cos{\theta})$ have a locally differentiable inversion?

$$f(r,\varphi,\theta)=(r\sin{\theta}\cos{\varphi},r\sin{\theta}\sin{\varphi},r\cos{\theta})$$ $$f:(0,\infty)\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}^3$$ How can I find out on which ...
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### Uniqueness of connected sum

Connected sum is defined as Wiki .But I think the result of connected sum is not unique. For example ,make connected sum on $S^2$ with itself . Then , the result can be $T^2$ or Klein bottle. Is it ...
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### Optimization: Via manifolds point of view of Lagrange multipliers method

My basis on differential manifolds calculus and differential geometry being very superficial, I'm trying to understand this section on WP's article. I'm not being able to realize why most of the ...
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### Open subset of $\mathbb {CP}^n$? [closed]

As picture below, how to show the $U_i$ is open subset of $\mathbb {CP}^n$ ?
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### Is $\mathbb R^2\backslash \{0\}$ a manifold?

Is $\mathbb R\backslash \{0\}$ a manifold ? Is $\mathbb R^2\backslash \{0\}$ a manifold ? I would say yes, but in the doubt, I prefer to ask.
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### Introduction to morse theory with applications to optimization

I am wondering if there are any easy-to-read introduction materials on morse theory (especially with applications to nonconvex optimization) for people with non-math background.
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### Divergent Curves and Complete Manifolds

I'm working on a problem in do Carmo's Riemannian geometry book (chapter 7, problem 5). He states that a divergent curve on a noncompact Riemannian manifold $M$ is a curve $\alpha: [0, \infty) \to M$ ...
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### Is the subgroup of homotopically trivial isometries a closed subgroup of the isometry group?

Let $(M,g)$ be a connected Riemannian manifold. Then according to the Steenrod-Myers-Theorem, the isometry group $\text{Isom}(M,g)$ of $(M,g)$ is a compact lie group with the compact-open topology. ...
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### Differential Forms on the Riemann Sphere

I am struggling with the following exercise of Rick Miranda's "Algebraic Curves and Riemann Surfaces" (page 111): Let $X$ be the Riemann Sphere with local coordinate $z$ in one chart and $w=1/z$ in ...
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### Relation between nonorientability of the Möbius strip and the Möbius bundle

There are two ways in which the open Möbius strip $M$ is related to orientability: $M$ is nonorientable as a manifold; $M$ is the total space of the nonorientable line bundle $M \to S^1$. Is there ...
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### Diameter of a Topological Manifold

I know that for a Riemannian Manifold is defined the concept of diameter. I wuold know if it's defined a similar concept for a most general Topological Manifold. Thanks in advance.
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### Does $S_1\subseteq \overline{S}_2-S_2$ $\implies$ $\dim S_1<\dim S_2$?

Question: Let $M$ be a smooth manifold and $S_1,S_2\subseteq M$ two locally-closed submanifolds (i.e. they are open in their closure). If $$S_1\subseteq\overline{S}_2-S_2,$$ is it true that \...
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### Obtaining embedding from geodesic

Suppose $M$ is an embedded sub-manifold of $D$-dimensional Euclidean space $E^D$, with embedding $\phi:M \hookrightarrow E^D$. And suppose I know the induced Riemmanian-metric $g$ on $M$, which ...
I am thinking about the intrinsic meaning (what this equation really means) about this equation. Suppose $\mathcal M$ is a smooth manifold embedded in $\mathcal R^d$, then for any $x \in \mathcal M$,...