# Tagged Questions

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### Decomposition of Laplacian into tangental and normal components w.r.t. submanifold

If I have the covariant Laplacian operator acting on a tensor e.g. $\nabla^2 h_{\mu\nu}$ on a (pseudo-Riemannian) manifold, and I have (say a codimension-2) submanifold, how can I "decompose" the ...
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### Euler characteristic for non-compact manifolds

How can one generalize the Euler characteristic to non-compact manifolds? Furthermore, is there a way to generalize the notion of an intersection number to non-compact manifolds, so that one could ...
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### Geometry of Commuting Hermitian Matrices

I am a physicist working on a project dedicated to the quantisation of commuting matrix models. In the appropriate formalism this problem is reduced to a quantisation in a curved space -- the space of ...
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### Prerequisite for Petersen's Riemannian Geometry

A professor recently told me that if I can cover the chapters on curvature in Petersen's Riemannian geometry book (linked here) within the next few months then I can work on something with him. ...
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### A few identities on differential forms

I just started studying differentiable manifolds, and I've encountered a few allegedly simple questions that aren't really simple to me. If possible, I'd be very happy to see full proofs (without ...
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### Nonorientable surfaces: genus or demigenus?

The genus $g$ of a closed, orientable surface is the maximum number of disjoint simple closed curves that can be drawn on the surface without disconnecting it. In terms of the Euler characteristic, ...
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### Lee, Introduction to Smooth Manifolds Solutions

Does anybody know where I could find the solutions to the exercises from the book Lee, Introduction to Smooth Manifolds? I use the freely available online version ...
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### Reference for introductory Lie Groups

I am currently learning about Lie groups,So kindly suggest a reference for Lie groups, which contains lecture on Manifolds as well.
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### Reference on Infinite Dimensional Manifold

I am studying manifold. For comprehension, I read the site http://en.wikipedia.org/wiki/Manifold, and there is some information about infinite dimensional manifold. Now I have two questions or ...
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### Transferring a result in PDE from open domain to a manifold

Can anyone recommend me something that in detail, talks about transferring a result in Sobolev space (as opposed to Holder spaces or something like that) that holds for open domains in $\mathbb{R}^n$ ...
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I am working on a subject of geometric group theory closely related to 3-manifolds, and in order to understand these links, I am seeking a good reference about 3-manifolds, as self-contained as ...
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### Grassmannian, Plucker coordinates

In which books can I find something about the grassmannian and the plucker coordinates ?
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### Any material on complexification?

These days, I met a problem on linear algebra: Suppose $A,B$ are real matrices. If there's a complex unitary matrix $U$ such that $U^*AU=B$, where $U^*=\overline U^\top$, namely, the conjugate ...
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### Curve in $\mathbb{P}^{n}(\mathbb{R})$, differentiable manifolds

I need a book which contains the demonstration that if $\Gamma$ is a curve in $\mathbb{P}^{n}(\mathbb{R})$ defined as $F(x_{1}, \cdots \ \ , x_{n+1}) = 0$ , with $F$ homogeneous polynomial, then ...
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Given a smooth surface $S \subseteq \mathbb{R}^3$, like the surface of sphere, we can define the following extended semimetric $d : S^2 \to [0, \infty]$, where $$d(x,y) = \inf\{\lVert x - p\rVert + ... 2answers 84 views ### Self-contained text on characteristic classes I am looking for a clear, self-contained text (either a book or lecture notes) that deals with characteristic classes, starting from the very basics (fiber bundle, principal bundle etc.), and ... 0answers 14 views ### Looking for a basic reference on propagators (in Topology) I am looking for a basic (preferably self-contained) reference where I can read about propagators (as they appear in Topology), and in particular Morse propagators. Thanks! 0answers 38 views ### Some questions on definition of Grassman manifolds In W.M. Boothby, An introduction to differentiable manifolds and Riemannian geometry , Page 64. We use coordinate correspondences \phi_j : U_j \to R^{k(n-k)} in order to define Grassman manifolds ... 0answers 67 views ### Definition of linking number for disjoint submanifolds of the sphere- A problem from Milnor's book In problem number 13 of Milnor's 'Topology from the Differentiable Viewpoint', the linking number for two compact boundary-less manifolds M,N \subset \mathbb{R}^{k+1} of dimensions m,n such that ... 1answer 187 views ### Applications of TQFTs beyond physics I'm giving a talk at a postgrad seminar on the topic of topological quantum field theories (TQFTs) with a mixed audience of pure and applied mathematicians. As such, I'd like to be able to offer some ... 0answers 40 views ### Analytic/Smooth/Continuous maps between a manifold and itself Let us suppose that M_{\omega} is a connected real-analytic manifold of dimension n. Then there is an associated smooth structure, \mathcal{C}^r structure (r non-negative integer) on it. Let ... 1answer 147 views ### Reconstructing a manifold from critical points I am teaching theoretical calculus this semester, and on the last discussion section we were discussing critical points of functions. I explained the idea of Morse theory, and a student of mine asked ... 1answer 60 views ### p-adic analytic group are closed subgroups of GL_n(\mathbb{Z}_p) for some n The article on pro-p-groups on Wikipedia tells us, that any p-adic analytic group can be found as a closed subgroup of GL_n(\mathbb{Z}_p) for some n \geq 0. Do you have a reference for that ... 1answer 289 views ### First proof of Poincaré Lemma I know that a way of proving Poincare lemma is to use the homotopy invariance and contractibility of the Euclidean space. Is there is a way of doing it directly (without using the contractibility of ... 5answers 168 views ### Books about manifolds? I would like to learn about manifolds. Please can someone recommend me a good book to learn about manifolds? 2answers 105 views ### Extending diffeomorphism to disk I am trying to prove the following If f:S^1 \to S^1 is a diffeomorphism it can be extended to a diffeomorphism F: D^2 \to D^2. But I can't seem to prove it. I proved it for homeomorphisms using ... 1answer 82 views ### If compact simply connected manifold has the same rational homotopy groups as S^n or \mathbb{C}P^n, must it have the same cohomology ring? The question came up while trying to shorten a paper I'm writing into submission-ready length. Let M be a compact simply connected manifold. By defininition, the rational homotopy groups of M ... 0answers 64 views ### A connected sum and wild cells Can we find such a connected sum of two spheres (in any dimension) that is not homeomorphic to the sphere? \def\R{\mathbb R} It seems that there should be examples like that, because there are lots ... 2answers 116 views ### whether gluing the faces of tetrahedron in pairs would get a manifold? I think gluing the faces of tetrahedron in pairs would get a manifold. Because gluing in pairs will make the result of gluing without boundary and there is not Y structure in the result. But it is ... 1answer 183 views ### Isotopy preserving inverse image f_t^{-1}(V) of a homotopy During a lecture I was given a bunch of easy propositions entitled as "observations" by the lecturer. But one of them seems more difficult and I have absolutely no idea how to "observe" it... I'd be ... 1answer 37 views ### Reference request for studying on space forms I would like to study on Space form, But I dont know what book or notes are suitable for beginning basically. Can someone help me? Thanks. 0answers 74 views ### A good reference for learning about super-differentiation & super-integration? I've looked at a couple of books for basic information for super-differentiation & super-integration - Rogers Supermanifolds, and Khrennikovs Superanalysis. Unfortunately both books lack a clear ... 1answer 187 views ### Manifold learning/nonlinear dimensionality reduction for beginners I'm a computer science graduate student. I recently discovered manifold learning. I think I understand the very basic, high-level concept of nonlinear dimensionality reduction, but I'd like a ... 3answers 510 views ### Asking for a good starting tutorial on differential geometry for engineering background student. I just jumped into a project related to an estimation algorithm. It needs to build measures between two distributions. I found a lot of papers in this field required a general idea from differential ... 2answers 214 views ### The Heisenberg manifold I am interested in the Heisenberg manifold, which is the quotient of the real Heisenberg group by the discrete Heisenberg (sub)group. It is a 3-manifold which may be viewed as a circle bundle over ... 1answer 79 views ### Theorem by Whitney For 0<k<\infty and any n-dimensional C^k manifold the maximal atlas contains a C^\infty atlas on the same underlying set by a theorem due to Whitney. Could someone please point me to ... 0answers 180 views ### References for basic level Differentiable Manifolds and Lie Groups I an undergraduate math student with a decent background in abstract algebra. I am looking forward to studying Lie groups this summer...I want some you to suggest good references for the following ... 0answers 23 views ### Condition for for Manifolds dual to functions What is the condition (locally compact/paracompact/Hausdorff/second countable?) for the following familiar statement: The category of (finite dimensional smooth) manifolds is contravariant ... 1answer 60 views ### Lebesgue covering dimension of a manifold I have found many sources saying that the Lebesgue covering dimension of a (topological or smooth) manifold is the same as the dimension of the manifold. Does anyone know where I can find the proof? 0answers 120 views ### Differentiable manifolds, Serge Lang I have started reading "Introduction to differentiable manifolds" by Serge Lang. In this book, Lang takes a different approach, by immediately introducing manifolds on arbitrary Banach spaces. His ... 1answer 128 views ### Figure \infty is immersion of circle Where can I find prove of: Figure \infty is immersion of circle. More thanks for a prove or a function between these manifolds. 0answers 48 views ### Standards in P.L. Topology About a week ago, the reading course on PL topology I'm going to follow started. The aim of the reading course is to understand the basics of PL topology and have a reasonable to good understanding of ... 1answer 237 views ### Infinite dimensional constant rank theorem Suppose you have an analytic map \phi : E \rightarrow \mathbb{C}^n, where E is a complex Banach space, and such that the rank of D \phi is constant. Is it true then that the set ... 1answer 56 views ### Metric Spaces needed for Differential Geometry I've asked here about some texts about differential geometry which doesn't assumes that the reader knows general topology. I've got good references as Do Carmo's Differential Geometry of Curves and ... 0answers 118 views ### Pullbacks as manifolds versus ones as topological spaces Let Y_1\overset{f}{\longrightarrow}X\overset{f_2}{\longleftarrow} Y_2 be smooth maps with a common target. Suppose that we have a pullback Z of the diagram in (Mfd). Questions: Suppose that we ... 1answer 184 views ### Green's function for the Yamabe problem I'm currently reading the paper on the Yamabe problem by Lee and Parker, and am looking for a reference for Theorem 2.8. Theorem 2.8 (Existence of the Green Function). Suppose M is a ... 1answer 83 views ### Hyperbolic spheres in the Poincare half-plane and fractional linaear transformations Let \mathbb{H} be the Poincare upper half-plane, seen as a Riemannian manifold with the metric$$\frac{dx^2+dy^2}{y^2}. Moreover, we consider the action of $\text{SL}_2(\mathbb{R})$ on ...
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I am looking to learn about manifolds for use in signal processing. I have a engineering degree where I have covered calculus and basic linear algebra, with this background in mind, does anyone have a ...