In my research work, I recently have come across a
system of three linear first order pde's whose characteristic polynomial has one
real and two complex conjugate zeros. I have searched the available
resources and could nowhere find out which category
(elliptic/hyperbolic/parabolic) it falls...
Being not an expert, my question might sound naive to students of mahematics. My question is how on earth a Lie group helps to solve an ode. Can anyone explain me in simple terms?
well:smile:
note carefully my figure 8: {(sin t, sin 2t): t ε (0, 2π)}. It is really not a 8. The middle portion is not a "cross". If t ε [0, 2π], then it is certainly not a manifold. By my "figure 8" IS a manifold and I am sure you know how. :)
Inverse will be continuous if you look at fig8 as a separate object not embedded in R^2. If you consider it to be embedded in R^2, then it will not be a homeomorphism.
map the nbd of π in (0, 2π) to one branch at the middle of fig8, right nbd of 0 to one portion of the remaining branch near the middle of fig8 and left nbd of 2π to the remaining portion of the remaining branch near the middle of fig8. Thats how u can cover the whole of fig8.
In stead of taking the subspace topology of ℝ^2, consider open intervals in (0,2π) to get mapped to the figure-8. This is indeed a homeomorphism, as in case of subspace topology of ℝ^2, it was not (because of the presence of the "cross" at the middle which was making our life hard).
What happened was I was reading a book where they have discussed the construction of charts for the figure-8: {(sin t, sin 2t): t ε (0,2π)}. Using the subspace topology of R^2, you cannot make it into a manifold. But there are topologies that admit its manifold structure, and there is a topology...
I haven't really tried to prove that there cannot be a single chart. But the usual charts that we have, like the 2 charts using stereographic projection, the 6 charts using hemispheres...and many others---all produce atleast 2 charts.
My question is about a general comment made in differential geometry books:
There cannot be a single chart for the manifold n-sphere.
I have been trying a proof for long but ...
Can you help me out?
can you give me some examples where we cannot define a connection?
I'm an engineering student. I asked these question regarding non-Riemannian description of defects in solids. In this field, there arises a manifold which is composed of disjoint non-compact parts and you cannot form a compact...
We all are familiar with the kind of differential geometry where some affine connection always exists to relate various tangent spaces distributed over the manifold, and from this connection two fundamental tensors, namely the Cartan's torsion and the Riemann-Christoffel curvature, arise.
Is it...
@ cgk
I think you are not aware of the topological classification of defects in ordered media. An ordered media is the generalization of crystalline materials. It can include liquid crystals to superfluids, ferromagnets and so on. There is a general classification scheme of defects occurring in...
Please help somebody on this problem...
When we topologically classify the defects in ordered media, we consider the character of the fundamental group of the associated order parameter space. To construct those groups, we circumscribe the line defects by circles and the point defects by...