# Questions tagged [manifolds]

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

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### Tu's An Introduction to Manifolds - Section 26.2 Cohomology of a circle, tabular form.

I'm trying to understand how to use the Mayer-Vietoris sequence to compute Cohomologies. There's a small chapter in Tu's Introduction to Manifolds explaining the basics, with some basic theory. More ...
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### Reason for defining $df(v)=D_vf$

I'm reading "A Visual Introduction to Differential Forms and Calculus on Manifolds", specifically the section on equivalence of directional derivatives to vectors acting as operators on functions ...
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### Showing that the differential is an immersion

If $f: X \rightarrow Y$ is an immersion of smooth manifolds, then show that $df: TX \rightarrow TY$ is also an immersion. The definition of immersion(when dim$X <$ dim$Y$) that I have is that for ...
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### Proving $R_{ij;m}=g^{kl} R_{ikjl;m}$.

In the coordinate $\{x^i\}$, the Riemann curvature tensor can be written as $$R=R_{ijkl}\,dx^i\otimes dx^j\otimes dx^k \otimes dx^l$$ and the Ricci curvature can be written as \text{Ric}=R_{...
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### Show that S is an embedded $k$-submanifold of $M$

Assume $M$ is a smooth manifold and $S$ is a subset of $M$ such that each point $p\in S$ has a neighborhood $U \subseteq M$ such that $S\cap U$ is an embedded $k$-submanifold of $U$. I want to show ...
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### Exact one form on non-simply-connected manifold

Let $M$ be a smooth $n-$dimensional manifold and $\alpha\in\Lambda^1(M)$ a smooth $1-$form. Let $N\subset M$ be an embedded submanifold where $\alpha|_N=0$. If $d\alpha=0$ on the whole $M$, can I say ...
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### Question about James Munkres's “Analysis on Manifolds”, p.112, Theorem 14.1

I'm an engineer who self-studies pure math in my free time, so I apologise in advance if this is a silly question. I am currently reading "Analysis on Manifolds" by James Munkres. Theorem 14.1 states ...
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### Subjects needed to learn Manifolds

The way I go about learning mathematics by my self is the following: I set up a goal, for example, the most recent one was "Complex analysis", and then I learn my way up to my goal, for example, what ...
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### Riemannian Exponential Map is a Homeomorphism outside the Cut Locus

Let $M$ be a connected and complete Riemannian manifold. The Hopf-Rinow Theorem guarantees that $Exp_p$, for any $p \in M$, is defined on all of $T_p(M)$. Further, this map is a diffeomorphism on a ...
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### Does Alexander duality hold for compact, orientable Homology sphere?

The Alexander duality Theorem is usually stated for a triangulable pair $(\mathbb S^n, Y)$ where $Y$ is a subset of the standard sphere $\mathbb S^n$. My question is: Does the duality also hold if we ...
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### Regarding equivalent definitions of Euclidean Submanifolds in Gallot, Hulin and Lafontaine's book Riemannian Geometry

In the book Riemannian Geometry by Gallot, Hulin and Lafontaine, a proposition which characterises equivalent definitions of submanifolds is given as follows: 1.3 Proposition The following are ...
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### Notation in differential forms

In a paper I've found a repeated notation with differential forms and, since I have never seen it before, my question is if there is a misprint or if it is an operation I don't know. I have a smooth ...
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### Levi-Civita connection from idempotents

Let $(M,g)$ be a closed Riemannian manifold. Let $V$ be a smooth complex vector bundle over $M$. We can write $V$ as the range of an idempotent $E$ in a matrix algebra $M_n(C^\infty(M))$ acting on a ...
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### Argument in a proof for scalar maximum principle

I'm trying to understand how an assertion made in the proof of the scalar maximum principle follows from the compactness of the manifold we're working with. The situation is as follows: I ...
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### On the geometric realization of a finite abstract simplicial complex which is connected, orientable $3$-manifold without boundary

Let $\Delta$ be an abstract simplicial complex on finitely many vertices and $|\Delta|$ be it's geometric realization. (https://en.m.wikipedia.org/wiki/Abstract_simplicial_complex) If $|\Delta|$ is ...
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### Second derivative test (and sign of laplacian at critical points) for manifolds

I'm trying to understand in more detail some of the justifications for a proof of the second derivative test for Riemannian manifolds, given below: I've never seen the Laplacian interpreted as an ...
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### Prove that the boundary orientation of $S^k = \partial B^{k+1}$ is the same as its preimage orientation

I would like to verify if my approach to this problem is the correct one or not. This problem is from "Differential topology" by Victor Guillemin and Allan Pollack . More specifically is the problem 3....