# 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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### Origins of Differential Geometry and the Notion of Manifold

The title can potentially lend itself to a very broad discussion, so I'll try to narrow this post down to a few specific questions. I've been studying differential geometry and manifold theory a ...
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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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### 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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### Why a Riemannian manifold minus one point is not complete? [closed]

Could you give me a proof that a Riemannian manifold minus one point is ever complete? Thanks!!
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### Integration of differential form on ellipsoidal surface with singularity in origin

As picture below ,I want to compute the (2) , because there is a singularity in $\{0\}$ and $\omega$ is closed . So ,I have $$\int_M\omega=\int _{\partial B_1(0)} \omega$$ I think there is a ...
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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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### 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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### 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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### 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 $x \in \mathcal M$,...
I need help understanding this proof in this book here: Concretely, I do not understand why it is okay to assume that $S$ can be parameterized by $n$ variables. Sure, it's an $n$-dimensional ...