Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I am confused about orientation and orientation reversing in local diffeomorphism $f$ from manifold $X$ to $Y$ at some points. So, what does $f$ orientation-reversing at a point mean?

share|improve this question

2 Answers 2

$X$ and $Y$ are oriented, which means every tangent space $T_xX$ and $T_yY$ has an orientation. $f$ is orientation-preserving (-reversing) at $x$ if and only if the linear map $df_x\colon T_xX\to T_{f(x)}Y$ preserves (reverses) orientation.

share|improve this answer

To elaborate on Ted's answer, to say that $X$ and $Y$ are oriented manifolds is to impose on the tangent spaces $T_x(X)$ and $T_{f(x)}(Y)$ a fixed, smooth choice of orientation at any given $x \in X$ (recall that an orientation of, say, $T_x(X)$ is just a choice of ordered basis $\alpha_1, \ldots, \alpha_n$, and likewise for $T_{f(x)}(X)$).

Now, $f$ is a local diffeomorphism at $x$, so the differential $df_x: T_x(X) \rightarrow T_{f(x)}(Y)$ is an isomorphism of vector spaces, and hence must take a positively-oriented basis $\alpha_1, \ldots, \alpha_n$ for $T_x(X)$ to a basis $B' = \beta_1', \ldots, \beta_n'$ for $T_{f(x)}(Y)$.

Since $Y$ is oriented, we already have a positively oriented $B = \beta_1, \ldots, \beta_n$ on $T_{f(x)}(Y)$. The matrix corresponding to the linear automorphism of $T_{f(x)}(Y)$ taking the ordered basis $B$ to $B'$ then has either positive or negative determinant. If the determinant of this matrix is positive, $B'$ is also a positively-oriented basis for $T_{f(x)}(Y)$, and we say that $f$ is orientation-preserving; if the determinant is negative, $B'$ is a negatively-oriented basis, and we say that $f'$ is orientation-reversing.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.