# 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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### Product of smooth maps from $M \to \mathbb{R}$ is smooth

In Lee's Introduction to Smooth Manifolds there is a exercise to proof that if $f: M \to \mathbb{R}$ and $g: M \to \mathbb{R}$ are smooth, so is $fg$. The question I'm having is the following: By ...
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### When is the inverse image of continuous function a submanifold

Let $f : R^n → R$ be a continuous function. Let $c ∈ Im f$. When is $f ^{−1}(c)$ a manifold? I know that if $f$ is a smooth function then $f ^{−1}(c)$ is a manifold if $c$ is a regular value. But ...
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### $M$ orientable implies $H_{n-1}(M, \mathbb{Z})$ is free Abelian group. [closed]

Let $M$ be a compact connected $n$-manifold without boundary, where $n \ge 2$. How do I see that if $M$ is orientable, then $H_{n-1}(M, \mathbb{Z})$ is a free Abelian group?
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### Understanding the definition of submanifold

I have defined a submanifold as: $M \subset \mathbb{R^n}$ is called a $k$-dim submanifold of $\mathbb{R^n}$ if $\forall x \in M$ the following condition holds: $\exists U \subset \mathbb{R^n}$ open, ...
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### How to prove that the half cone doesn't have a tangent space at its vertex?

I don't know how to prove that the half cone (including its vertex) $\lbrace (x,y,z): x^2+y^2=z^2 , z\geq 0 \rbrace$ does not have a tangent space at its vertex (0,0,0).
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### Barycentric subdivision of regular CW decomposition is a combinatorial manifold?

Suppose $X$ is a PL manifold (with boundary) and let $(X,X_{i})$ be a regular CW complex. Is the barycentric subdivision of $(X,X_{i})$ a combinatorial manifold? Answer given in comments. Definition: ...
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### Duality between tangent and cotangent bundles

Given a smooth manifold $M$, the cotangent bundle $T^*M$ is dual to the tangent bundle $TM$ "fiberwise", i.e. $\forall x\in M$, $T^*_x(M)=(T_x(M))^*$. Now, if the manifold is a vector space, then the ...
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### How to characterize the dimension of a manifold using homology?

This might be a trivial question but I'm a physicist, not a mathematician. For me, the n dimensional euclidean space is n dimensional as a vector space. I have heard however that there more intrinsic ...
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### Lifting of triangulation

In "Complex Analysis 2: Riemann Surfaces, Several Complex Variables, Abelian Functions, Higher Modular Functions" and many other books is described a lifting of triangulations for branched covers ...
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### Is every manifold a metric space?

I'm trying to learn some topology as a hobby, and my understanding is that all manifolds are examples of topological spaces. Similarly, all metric spaces are also examples of topological spaces. I ...
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### The cone is not a regular submanifold of $\mathbb{R}^3$

I am not very familiar to differentiable manifolds, so I would appreciate some hints or reasonings about why the cone $$M = \{(x,y,z)\in\mathbb R^3:x^2+y^2-z^2 = 0, z\geq0\}$$ is not a regular ...
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### Contraction of $(2k,2l)$tensor

In picture below ,how to get the equality in red box?
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### A beginner's question of Riemannian Geometry.

In picture below ,I don't know why $\Phi^{-1}(F)=(F(\phi^i,e_j))_{i,j=1}^n$
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### Example of fibre bundle is locally product but not globally

When I read the below picture ,I can't make a example for claim of red box.
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### Can someone help me understand the Euclidean metric?

A Euclidean metric is defined as: $g_{euclid} = g_{ij} = \delta_{ij} dx^i \times dx^j = dx^1dx^1 + \ldots + dx^ndx^n$ Can someone explain the following: why do we use $dx^i$ instead of $x^i$ which ...
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### Visualization of SU(3)

I am trying to visualize the $SU(3)$ group used in quantum field theory. I have a (reasonably) good understanding of $SU(2)$ as the double cover of $SO(3)$ and also that this is homeomorphic to $S^3$. ...
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### Understanding the differentiable structures on Grassmann manifold

I am reading in the book Differential Analysis on Complex Manifolds by Raymond. I have a trouble in understanding the differentiable structures on Grassamann space. I uploaded the picture of the page ...
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### Topological Boundary vs Manifold Boundary

Let $A$ be the open unit disc in $\mathbb{R}^2$ and $B$ be the closed unit disc in $\mathbb{R}^2$. The toplogical boundary of $A$ and $B$ is $S^1$. This I understand. The manifold boundary of $A$ ...
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### The composition of a smooth map with a linear isomorphism is smooth

I'm reading Jeffrey Lee's Manifolds and Differential Geometry section on manifolds with boundary. In page 50, given an $n$-dimensional manifold $M$, he builds a smooth atlas for $\partial M$ in the ...
I am currently reading Milnor's paper which discusses the group action on spheres without fixed point. At the second page of the paper, he denotes $$M^n*M^n$$ to be a symmetric product of a manifold. ...