Are the chern classes of the sphere bundle from a complex line bundle the same?

Question

The sphere bundle of a complex line bundle $$ L \to M $$ is an $S^1$-bundle over $M$. Moreover, since complex vector bundles are always orientable, we have that the induced $S^1$-bundle is principal.

Since Chern-Weil theory gives us a way to construct chern classes for principal bundles, does the chern class of this $U(1)$-bundle agree with the chern class for $L$?

Answer 1

Yes. If $P_L$ is the principal $S^1$-bundle associated to $L$ then there is an isomorphism of line bundles $P_L\times_{S^1}\mathbb{C}\cong L$ over $X$ by definition. The Chern classes are natural under pullback so both the bundles have the same Chern classes.