In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or n-manifold for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of n-dimensional Euclidean space.
One-dimensional manifolds include lines and circles, but not figure eights. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane.
The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. The concept has applications in computer-graphics and augmented-reality given the need to associate pictures (texture) to coordinates (e.g. CT scans).
Manifolds can be equipped with additional structure. One important class of manifolds are differentiable manifolds; their differentiable structure allows calculus to be done. A Riemannian metric on a manifold allows distances and angles to be measured. Symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics, while four-dimensional Lorentzian manifolds model spacetime in general relativity.
The study of manifolds requires working knowledge of calculus and topology.
Homework Statement
Let ##A## be open in ##\mathbb{R}^n##; let ##\omega## be a k-1 form in ##A##. Given ##v_1,...,v_k \in \mathbb{R}^n##, define
##h(x) = d\omega(x)((x;v_1),...,(x;v_k)),##
##g_j(x) = \omega (x)((x;v_1),...,\widehat{(x;v_j)},...,(x;v_k)),##
where ##\hat{a}## means that the...
Hi all,
I'm quite confused concerning the definition of tangent vectors and tangent spaces as presented in Munkres's Analysis on Manifolds. Here is the book's definition:
Given ##\textbf{x} \in \mathbb{R}^n##, we define a tangent vector to ##\mathbb{R}^n## at ##\textbf{x}## to be a pair...
Homework Statement
Let ##O(3)## denote the set of all orthogonal 3 by 3 matrices, considered as a subspace of ##\mathbb{R}^9##.
(a) Define a ##C^\infty## ##f:\mathbb{R}^9 \rightarrow \mathbb{R}^6## such that ##O(3)## is the solution set of the equation ##f(x) = 0##.
(b) Show that ##O(3)## is a...
Homework Statement
Let M and N be orientable m- and n-manifolds, respectively. Prove that their product is an orientable (m+n)-manifold.
Homework Equations
An m-manifold M is orientable iff it has a nowhere vanishing m-form.
The Attempt at a Solution
I assume I would take nowhere...
I am looking for books that introduce the fundamentals
of topology or manifolds. Not looking for proofs and rigor.
Something that steps through fundamental theorems in the
field, but gives conceptual explanations.
Dear Folks:
In most textbooks on differential geometry, the regular theorem states for manifolds without boundaries: the preimage of a regular value is a imbedding submanifold. What about the monifolds with boundaries...
Homework Statement
This problem is in Analysis on Manifolds by Munkres in section 25. R means the reals
Suppose M \subset R^m and N \subset R^n be compact manifolds and let f: M \rightarrow R and g: N \rightarrow R be continuous functions.
Show that \int_{M \times N} fg = [\int_M f] [...
Ok, so I don't have much of an intuition for frame bundles, so I have some basic questions.
A frame bundle over a manifold M is a principle bundle who's fibers are the sets of ordered bases for the vector fields on M right.
1) This means that any point in the fiber (say, over a point m in M)...
hi!
i would very much like to have the "Lectures on Comlex Manifolds" by Philip Candelas. It was recommended by an instructor of a course on complex geometry i took some time ago, but sadly its out of print and not in our library. Does anyone has this documents in electronic form?
When reading other threads, following question crept into my mind:
When given a manifold, why shouldn't I give it distance function by giving it a simple metric function, that is MxM→ℝ with the usual axioms? I could happily measure distances in coordinate-independent way for ever after...
Hi everebody,
I want to clear something.An n-dimentional differential manifoled is locally endowed by topologies defined by the metrices from the local parametrisations.I suppose that these topologies may all be different.Am i right?If i am mistaken ,then why?
thank's
I've always assumed that for a non-Euclidean manifold to exist, it has to be ambient in some higher-dimensional Euclidean space, like how a 2-sphere is ambient in 3-dimensional Euclidean space. But I've been hearing hints that higher-dimensional embedding is in fact unnecessary to define a...
If you have a strip and you bring it around so that the ends join, that is a manifold, call it X for convenience. If instead, you put a single twist in it before joining the ends, that is a Mobius strip, which is not homeomorphic to X. If you instead put two twists in it before you join the...
Hi, All:
AFAIK, every complex manifold can be given a symplectic structure, by using
w:=dz/\dz^ , where dz^ is the conjugate of dz, i.e., this form is closed,
and symplectic. Still, I think the opposite is not true, i.e., not every symplectic
manifold can be given a complex...
Hi,
I want to ask a problem from Lee ' s book Introduction to Smooth Manifolds: Let F_p denote the subspace of C^\inf(M) consisting of smooth functions that vanish at p and let F^2_p be the subspace of F_p spanned by functions fg for some f,g \in F_p. Define a map \phi: F_p----->...
Hi,
I'm having trouble understanding why is tangent space at point p on a smooth manifold, not embedded in any ambient euclidean sapce, has to be defined as, for example, set of all directional derivatives at that point.
To my understanding, the goal of defining tangent space is to provide...
If we consider a Riemannian surface as a one-dimensional complex manifold, what does that tell us about its intrinsic curvature? I mean for one-dimensional curves we know they only have extrinsic curvature so it depends on the embedding space, this doesn't seem to be the case for one-dimensional...
Hi!
I want to know if any smooth manifold in n-dimensional euclidean space can be compact or not.
If it is possible, then could you give me an example about that?
I also want to comfirm whether a cylinder having finite volume in 3-dimensional euclidean space can be a smooth manifold.
I...
Consider the infinite disjoint union M = \coprod\limits_{i = 1}^\infty {M_i },where M_i 's are all manifolds of finite type of the same dimension n.Then the de Rham cohomology is a direct product H^q (M) = \prod\limits_i {H^q (M_i )}(why?),but the compact cohomology is a direct sum H_c^q (M) =...
This is really just a general question of interest: can every smooth n-manifold be embedded (in R2n+1 say) so that it coincides with an affine variety over R? Does anyone know of any results on this?
I am having some trouble understanding the notion of an orientated manifold. But first let me get some preliminary definitions out of the way:
A diffeomorphism is said to be orientation-preserving if the determinant of its Jacobian is positive. A k-manifold M in \mathbb{R}^n is said to be...
Background:
I'm going to be a junior, having very strong Analysis and Algebra yearlong sequences, in addition to a very intense Topology class, and a graduate Dynamical Systems class.
For this coming Fall, I'm sort-of registered for this class, titled "Manifolds and Topology I" (part of a...
Right now, I'm reading through Lee's Intro to Smooth Manifolds and I was wondering if there is a website somewhere that has problems different from the book. Or if there is another book out there that covers about the same material as Lee's that would be good to.
Thanks in advance.
Hi, All:
I am trying to show that the connected sum of orientable manifolds M,M' is orientable , i.e., can be given an orientation. I am using the perspective from Simplicial Homology.
Consider the perspective of simplicial homology, for orientable manifolds M,M', glued about cycles C,C'...
Hi, All:
Given manifolds M,N (both embedded in $R^n$, intersecting each other transversally,
so that their intersection has dimension >=1 ( i.e. n -(Dim(M)-Dim(N)>1) is the intersection
a manifold?
Thanks.
I am currently going through some online notes on differential geometry and general relativity. So far I have been following pretty well until I got to 3.4 cont'd letter (e) of http://people.hofstra.edu/Stefan_Waner/diff_geom/Sec3.html" document and the definition directly after. My first...
My definition of diffeomorphism is a one-to-one mapping f:U->V, such that f and f^{-1} are both continuously differentiable. Now, how to prove that if f is a diffeomorphism between euclidean sets U and V, then U and V must be in spaces with equal dimension (using the implicit function theorem)?
Quoted from a book I'm reading:
if f is any function defined on a manifold M with values in Banach space, then f is differentiable if and only if it is differentiable as a map of manifolds.
what does it mean by 'differentiable as a map of manifolds'?
im trying to read calculus on manifolds by michael spivak and am having a tough time with it. if anyone could recommend a more accessible book (perhaps one with solved problems) id really appreciate it.
What yould you answer if a professor asks you,
Why are the classification theorems of manifolds so important? Why was the classification of surfaces celebrated?
Hi everyone, been away for a while I got bogged down with my classes so didn't have time to work on this book and haven't been on the forums much. Was getting caught back up to where I was before in here and I ran into a problem that I can't figure out the notation on.
I am only looking for...
hi friends :)
is there someone who has studied the spectrum of a Riemannian Laplacian? I have a question on this subject. Thank you very much for answering me.
A little embarrassing, but I have had very little exposure to anything involving manifolds and am trying to work through these notes over spring break. I will have many questions on even the simplest concepts. In this thread I hope to outline these as I encounter them, and if anyone can help I...
Homework Statement
Let \phi \in C^{\infty}_{0}(\mathbb{R}^2) and f: \mathbb{R}^2 \to \mathbb{R} a smooth, non-negative function. For c > 0, let < F_c, \phi > := \int_{\{f(x,y) \le c\}} \phi(x,y)\mbox{dx dy} . Supposing the gradient of \frac{\partial f}{\partial x} is nonzero everywhere on M...
my book defines an orientation preserving parametrization of a manifold as one such that:
Ω(D1γ(u), ..., Dkγ(u)) = +1 for all u in the domain of γ, where D1,...Dk are the derivatives of the parametrization γ.
my book also defines the orientation of a surface in R^3 by Ω(v1,v2) = sgn...
Homework Statement
Taken from Wiki:
a manifold is a mathematical space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold. Thus, a line and a circle are one-dimensional manifolds, a plane and sphere (the surface of a ball)...
Homework Statement
OK I have a Differential Calculus exam next week and I do not understand about Differential Manifolds.
We have been given some questions to practise, but I have no idea how to do them, past a certain point.
For example
1. Study if the following system defines a manifold...
I understand that symplectic manifolds are phase spaces in classical mechanics, I just don't understand why we would use them. I understand both the mathematics and the physics here, it is the connection between these areas that is cloudy...
What on Earth does the symplectic form have to do...
Hello,
I am learning about manifolds but I am not understanding part of the definition. This is what I'm looking at for defining the n-manifold M.
(i) M is Hausdorff
(ii) M is locally Euclidean of dimension n, and
(iii) M has a countable basis of open sets
I have a problem with (ii)...
Hi all,
I was reading a paper written by Brian Greene sometime ago on flop
transitions where one can essentially change the topology of the
manifold but the four-dimensional physics that applied to the older
manifold still holds. From that I am trying to extrapolate the
following: Is it...
Hello there!
I just started reading Topological manifolds by John Lee and got one questions regarding the material.
I am thankful for any advice or answer!
The criteria for being a topological manifold is that the space is second countantable ( = there exists a countable neighborhood...
Hi all,
I am rather new here but I think I am posting in the right place, if not my apologies :) Alright so I've been working on calibi-yau manifolds for some time now and string theory (let's take a generic approach here and not worry about which version) states that the shape of the manifold...
In the notes attached in this thread:
https://www.physicsforums.com/showthread.php?t=457123
How do we go about the exercise at the bottom of p67/top of p68?
And secondly, at the top of p69, he giveas the example and invites us to check that (\phi^* g)_{\mu \nu} = diag ( \sin^2{\theta})...
Hi all,
This may seem like a simple problem, but I just want to clarify something.
The issue is the relationship between sub-Riemannian manifolds and cometrics. In particular, say we have a manifold M and a cometric on the cotangent bundle T*M. Firstly, it is my understanding that somehow a...
I saw a nice formulation of the variation on odd dimensional manifolds in the paper of http://arxiv.org/abs/math-ph/0401046" :
The referenced book of Arnold uses completely different formalism than this.
I don't see clearly the connection between the traditional calculus of variations...
Hello,
I read that when a manifold has a flat metric, the geodesics are always straight lines in the parameter space. I have two questions:
(1)
If we are given a Clifford torus S^1 \times S^1 (which is flat), how do we compute the geodesic between two points? Is the following correct...
I hope this is not the wrong place to ask this...
Can anybody tell me if it is possible to find "Spivak calculus on manifolds" on line (a PDF copy for example)
Thanks