# Search results

1. ### Problems wiith the notation

Hello, i have found a notation, which i never saw before and i can't give some reasonable definition for it. Perhaps some of you guys have seen it before and can help me. let k \in \mathbb{Z}^n and N be a constant, what does <k>^N mean? I will put more of the context, where this...
2. ### Orientation of submanifold

Thank you for your help. Let \omega be our volume form. , \omega :M->\Lambda^m M That is \omega assigns to each point a alternating tensor. And we need a smooth vector field X:M->TM. But i don't know how \omega induces a vector field. \omega(p) \in \Lambda^m (T_p M) This is a tensor...
3. ### Orientation of submanifold

Hello, yes you are right, X is a smooth vector field on M. We have a few equivalent definitions of "orientable". A manifold is orientable if 1) det d(f \circ g^{-1})>0 , \forall f,g whereas f, g are coordinate maps of the manifold. <=>2) we have a non-vanishing differential m-form...
4. ### Orientation of submanifold

Hello, here is my exercise: Let M be a orientable manifold of dimension m and let N be a submanifold of M of codimension 1. Show that N is orientable <=> it exists a X \in \tau_1 (M), s.t. span<X(p)> \oplus T_p N= T_p M \; \forall p\in N The X is a vector field, i.e. X(p) is an...
5. ### Why has to be M,N be orientable?

Oh i think it is a misunderstanding. You are right, we inly need a single atlas, which is oriented, to show that a manifold is oriented. But I want to show something else. i want to show tha MxN is not orientable, if M is not. That is i have to show that any atlas is not orientable of MxN! I...
6. ### Why has to be M,N be orientable?

Yes of course there is! But not any atlas has to be in that form! This is a real problem. For example, if (U_i,f_i) is a atlas for M and (V_i,g_i) is a atlas for N => (U_i x V_i, f_i x g_i) is a atlas for MxN. But it has not to be in this form. That is f,g can depend on both, the element in...
7. ### Why has to be M,N be orientable?

Hello, There are different äquivalent criteria for a manifold to be orianted. A manifold is oriented if it has an atlas, s.d. the differential of the coordinate changes have positiv determinant. Therefore the product MxN has a canonically atlas, which also satisfy this criterion. I hope...
8. ### Why has to be M,N be orientable?

Hello, I want to solve this Problem: If M,N are manifolds then MxN is orientable iff M,N is orientable. I have solved the direction "<=" This was no problem. But i have a lot of problems to solve the other direction! Let us assume that MxN are orientable. Why has to be M,N be...
9. ### Construction of an atlas

Hello, I want to show the following little exercise: If we have a manifold M and a atlas A of M, s.t. for all coordinate maps x,y \in A: det \; (d(x\circ y^{-1})<0). Then there is a atlas A' s.t. for all x',y': (det \; d(x\circ y'^{-1})>0) I try to change the coordinate maps by...
10. ### Algebra Multiplikation

Can nobody help me?:-( It would be very nice, if someone can help me. Thanks in advance.
11. ### Algebra Multiplikation

Ok, thank you! Do you know, how i can construct a isomorphism? Is this a Isomorphism: \phi: \bigotimes_{i=1}^\infty L(V_i) \rightarrow L(\bigotimes_{i=1}^\infty V_i), defined by x=\bigotimes_{i=1}^\infty (f_i) \to \phi(x): \bigotimes_{i=1}^\infty V_i \to \mathbb{K}, \otimes e_i \to \pi...
12. ### Algebra Multiplikation

Hello, I want to show that the Algebras L(\bigotimes_{i=0}^\infty V_i)\; and\; \bigotimes_{i=1}^\infty \; \L (V_i) are isomorphic! But for this i need to know the algebra-structure on \bigotimes_{i=1}^\infty \; \L (V_i). How the multiplication is defined on this space? Regards
13. ### Differential form

Hello, thank you, but why it is a differential form? We have defined differential form as a smooth section of the projection map. Therefore i have to show this. But for instance i do not see why it is smooth. Regards
14. ### Differential form

Hello, I try to understand differential forms. For istance i want to prove that h=e_1\wedge e_2 + e_3\wedge e_4 is a differential form, where e_1,..,e_4 are elements of my basis. Do you have a idea, why this is a differential form? Regards
15. ### Decomposable tensors

Hello, Thank you for your help! are these examples correct?: For dimV=2 We have \Lambda(V) = IK \times T^1(V) \times T^2(V). -Then a homogeneous elm. would be all elm. in v\in \IK or v\in T^1(V) or v\in T^2(V). Where T is the Alternator. -Is it allowed to use the single exterior...
16. ### Decomposable tensors

Problem: V a vector space with dimV \le 3, then every homogeneous element in \Lambda(V) is decomposable. So, this exercise doesn't sound very difficult. My problem is, that i don't know the definition of homogeneous and decomposable. Can you please help me? Thank you