1
vote
1answer
68 views

Homework help: Vector functions

i need help with an assignment: We have 2 vector functions, described below (Its a screenshot from my mathcad document, cant format it here) I need to calculate at what "t" value the 2 functions ...
0
votes
1answer
62 views

When is the restriction of a Lorentzian metric to a regular submanifold degenerate everywhere?

Let $M$ be a $C^\infty$ manifold, $N\subset M$ be a regular submanifold and $g$ be a Lorentzian metric on $M$. I would like to find $M$, $N$, $g$ such that the restriction of $g$ to the tangent ...
4
votes
1answer
194 views

Computing Chern Classes of Tautological Line Bundles

I cannot find any references with how to handle this tautological line bundle $V \rightarrow P(\mathbb{H}^2)$ where $\mathbb{H}$ are the Hamilton quaternions. I know it is a complex rank 2 vector ...
1
vote
1answer
53 views

Change of frame of a vector bundle

Let $(E, \pi, M)$ be a complex vector bundle of rank $k$. Let $U \subset M$ be an open set and let $f = (s_1, \dots, s_k)$ be a frame (i.e. s_i are linearly independent). Of course $U$ is supposed ...
7
votes
2answers
214 views

Dual of a holomorphic vector bundle

Let $(E,\pi,M)$ be a holomorphic bundle, i.e. $(M,J)$ is a complex manifold and $\pi \colon E \to M$ is a complex bundle such that there exists a trivialization with holomorphic transition functions. ...
2
votes
1answer
129 views

Prove that a tensor field is of type (1,2)

Let $J\in\operatorname{end}(TM)=\Gamma(TM\otimes T^*M)$ with $J^2=-\operatorname{id}$ and for $X,Y\in TM$, let $$N(X,Y):=[JX,JY]-J\big([JX,Y]-[X,JY]\big)-[X,Y].$$ Prove that $N$ is a tensor field of ...
2
votes
1answer
214 views

Triviality/non-triviality of line/circle bundle over $S^3$

I am interested in whether I can find non-trivial bundles in the case of a fiber bundle with base space $S^3$ and fiber either $\mathbb{R}$ or $S^1$. I know that in the case of a principal bundle ...
0
votes
1answer
92 views

What is the isomorphism between E* and Hom(E, M$\times$R)

What is the isomorphism between dual vector bundle $E^*$ and $\mathrm{Hom}(E,M\times \mathbb R)$? There is a natural isomorphism on bundle that is $\mathrm{Hom}(E,E')=E^*\otimes E$, therefore I am ...
1
vote
1answer
206 views

If bundle map is a local isomorphism, then bundle map is diffeomorphic?

Suppose $\phi$ is a bundle map from $E$ to $F$, where $E$ and $F$ are two vector bundles over a manifold $M$. Now for each $x\in M$, $\phi$ restricted to $E_x:E_x\to F_x$ is a isomorphism, how to show ...