# 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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### $\mathbb{CP}^2$ realizable as boundary of compact smooth $5$-manifold? [duplicate]

As the title says, can $\mathbb{CP}^2$ be realized as the boundary of a compact smooth $5$-manifold?
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### What does $C^k$ at a single point mean?

I'm reading textbooks on manifolds. I usually see a saying that "function $f$ is $C^k$ at a point $p \in M$". I am confused with this. What does this mean? Furthermore if $f$ is $C^k$ at $p$, is it ...
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### What is the motivation for wanting to remove redundant terms that arise from the wedge product of two multilinear functions?

I am currently working through Loring Tu's "An Introduction to Manifolds," specifically the section in which the exterior algebra of multicovectors is introduced, and I am having trouble grasping the ...
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### Poincaré Duality in Middle Dimension

I am reading a paper that states the following theorem without proof: Poincaré duality in middle dimension: Let $M$ be a connected oriented manifold of even dimension $2d$. Then the cup product ...
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### Noncanonical isomorphism of spaces of differential forms

Let $\pi: V \to M$ be a smooth $n$-dimensional vector bundle over $M$. Are the spaces of differential forms $\Omega^i(V)$, $\Omega^i(V^*)$ noncanonically isomorphic? If so, how do I see this? Is there ...
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### Left-invariant vector fields on the circle $S^1$

I'm trying to find the left-invariant vector fields on the circle $S^1$. If I understand correctly, $S^1$ is given the group structure of the multiplicative group of complex numbers on the unit ...
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### compactification of $\Bbb R^n$

I want to show that $n$ dimensional real projection space is compactification of $\Bbb R^n$, how can I do that? I can show PR^n is compact, and R^n is localy compact, but I have problem to show R^n ...
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### Derivative of map $f: S^n \to \mathbb{R}P^n$ is an isomorphism

I'm trying to show that the map $f: S^n \to \mathbb{R}P^n$ given by sending a unit vector $x$ in $S^n \subset \mathbb{R}^{n+1}$ to the line spanned by $x$ in $\mathbb{R}P^n$ has injective derivative. ...
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I have a question about tensors and metrics: Let $M=\{(t,x,y,z)\in \mathbb{R}^4: t>-1 \}$ and let $g=(1+t)dtdx+dy^2+dz^2$ Show that g is a metric on $M$. I did the next, I have the basis $\{ \... 1answer 33 views ### On computing the Differential of a Smooth Map In class, we proved Jacobi's formula for the differential of the determinant using the following formula for the differential of a smooth map$F$between manifolds$M$and$N$$$D_A F(B) = \frac{\... 1answer 56 views ### Why is \phi^* g = g a PDE for a pseudo-Riemannian metric g on a manifold? Given a (locally trivial) bundle \pi: E \to M a PDE of order k is usually defined to be a submanifold of the jet-bundel J^k(E). Now assume E = M \times M and \pi is the projection on the ... 1answer 47 views ### What is the skew-symmetric part of the covariant derivative of a one-form? This is a followup question to here. Let E \to M be a vector bundle with connection D := \nabla. Extend D to E^* and \text{Hom}(E, E). Let E = TM here, and suppose that the torsion is 0:... 0answers 69 views ### Which homology groups of a closed orientable 6-manifold can be isomorphic to \mathbb{Z}^3? List all i for which there is a closed orientable 6-manifold M with H_i(M) =\mathbb{Z}\oplus\mathbb{Z}\oplus\mathbb{Z} I am working on an old exam problem and this one stumped me. Progress ... 1answer 83 views ### Why is the image of the implicit function in the implicit function theorem not open? We have a continuously differentiable function f from \mathbb{R}^{n+m} to \mathbb{R}^n, and we find a continuously differentiable function g which maps points from \mathbb{R}^m into \mathbb{... 1answer 38 views ### Difference between Grassmann and Projective space? I recently read the first few chapters of Pyramid Algorithms: A Dynamic Programming Approach to Curves and Surfaces for Geometric Modeling and I am confused about ... 1answer 61 views ### Question about connections on the dual bundle. Let E \to M be a vector bundle with connection \nabla. Extend \nabla to E^* and E^* \otimes E in the regular fashion. Is \text{Id} \in E^* \otimes E necessarily parallel? 0answers 23 views ### Quadrature over a (smooth, compact, convex, etc.) Riemannian manifold Problem setting Consider three points on the surface of the earth (which I want to assume to be a perfect ellipsoid here) that are pairwise sufficiently close for unique geodesics to be found between ... 0answers 32 views ### When and where to check the formal definition of a manifold In most texts on GR we are first introduced to a formal and rigorous definition of a manifold. We then learn the point that in GR "any coordinate system" might be used for the 4D spacetime metric. ... 1answer 31 views ### Reference request for stochastic processes on manifolds I'm looking for some references on stochastic processes on manifolds. The more introductory the better. Thanks. 1answer 52 views ### Every \mathcal{C}^1 manifold can be made smooth? I heard of a theorem saying that each \mathcal{C}^k-manifold with k\geq 1 can be made into a smooth manifold, i.e. \mathcal{C}^{\infty} (by restriction of the atlas). However, I cannot find ... 1answer 33 views ### Lemma characterizing second fundamental form, do not understand step Consider an excerpt of a lemma and part of its proof from a Riemannian geometry textbook. Lemma. The second fundamental form is independent of the extensions of X and Y; bilinear ... 1answer 47 views ### Quotient by a discrete subgroup of a Lie group I was reading Fulton Harris' Representation theory, A first course, where I came across the following: Let H be a Lie group and T be a discrete subgroup of its center Z(H). Then there exists ... 1answer 30 views ### Understanding Hempel's proof of uniqueness of cube with handles In Hempel's 3-Manifolds book, Theorem 2.2 says that if P and Q are two cubes, both with n handles, and both are orientable, then they are homeomorphic. He defines a cube with handles as a 3-... 1answer 61 views ### Are the collections {rings}, {smooth manifolds} larger or smaller than {ab. groups}, {top. manifolds}? Intuitively, the collection of smooth manifolds feels smaller than that of topological manifolds: they are not just locally nice continuous objects, but even smooth. Similarly, when one finds that an ... 1answer 30 views ### Is every convex cone a manifold? Let C \subseteq \mathbb{R}^n\setminus \{0 \} be a connected convex cone*. Question: Is C always a topological manifold (perhaps with boundary)? A smooth one? Does anything change if we do not ... 1answer 36 views ### Vector bundle vs Total Space On page 59 in Lee's "An Introduction to Smooth Manifolds" the author writes, "Let E be a smooth vector bundle over a smooth manifold M, with projection \pi:E\to M." I thought the vector bundle ... 0answers 31 views ### Does anything obstruct Mostow-Prasad rigidity for orbifolds? If we phrase the Mostow-Prasad rigidity theorem algebraically, it goes like this (let \mathcal{H}^n be a model for hyperbolic n-space). For n>2: if \Gamma,\Delta<\mathrm{Isom}(\... 2answers 32 views ### Choice of chart and altas Manifold is second countable space. Should charts in altas also be countable? I don't think it has to be. But some how the second countable condition enforces me to think like that.. 0answers 34 views ### Is T^2 \# \mathbb RP^2\cong \mathbb RP^2\# \mathbb RP^2\# \mathbb RP^2? I was reading a little about how to imagine the projective plane and I have some weird intuition that says T^2 \# \mathbb RP^2\cong \mathbb RP^2\# \mathbb RP^2\# \mathbb RP^2. Is this true, and if ... 2answers 59 views ### Representable homology classes on smooth manifolds Let X be a closed (compact without boundary) smooth manifold. We can consider its singular homology H_*(X,\mathbb{Z}). Let H_{k}(X,\mathbb{Z}) be the k-th singular homology group of X and ... 0answers 24 views ### Poincaré-Hopf Index Theorem - Intuitive explanation I heard about an interesting theorem. Poincaré-Hopf Index Theorem : If \vec{v} is a smooth vector filed on the compact, oriented manifold X with only finitely many zeros, then the global ... 1answer 35 views ### Tensor field acting on vector fields and covector fields gives a function? If I have a (p,q) tensor field that acts on p covector fields and q vector fields then does T\left(X_1,\dots,X_p,Y_1,\dots,Y_q\right) return a function f defined on the manifold by$$f\left(... 1answer 71 views ### The classifying space of open covers of a manifold Let$M$be a manifold of dimension$d$and let$\mathsf{Disk}_{/M}$be the category of open subsets of$M$that are diffeomorphic to$\mathbb{R}^d$with morphisms given by inclusions. Let$\mathrm{B} ...
I am asked to prove that for a surface in $\mathbb{R}^3$ with local coordinates in a chart, u,v, the coefficients of the second fundamental form can be calculated as follows: eg.: (first entry in II) ...