# Orientability and unit tangent bundle of surfaces

What can we say about the orientability of the unit tangent bundle $$UTM$$ of $$M$$?

The unit tangent bundle of the sphere $$S^2$$ is $$\mathbb{R}P^3$$ see A question on the unit tangent bundle of the sphere and $SO(3)$

The unit tangent bundle of $$T^2$$ is $$T^3$$ (torus is Lie group so parallelizable so unit tangent bundle is trivial)

The unit tangent bundle of an orientable surface $$\Sigma_g$$ of genus $$g \geq 2$$ is $$UT(\Sigma_g) \cong SL_2(\mathbb{R})/\pi_1(\Sigma_g)$$

For the non orientable surfaces the story is similar. Every unit tangent bundle is double covered by the unit tangent bundle of its orientable double cover.

For example the unit tangent bundle of the projective plane is the homogeneous (q=1) lens space with fundamental group $$C_4$$ cyclic of order 4. $$UT(\mathbb{R}P^2) \cong L_{4,1} \cong SU_2/C_4$$ So in particular $$UT(\mathbb{R}P^2)$$ is orientable. See https://www.projecteuclid.org/journals/nihonkai-mathematical-journal/volume-13/issue-1/Unit-Tangent-Bundle-over-Two-Dimensional-Real-Projective-Space/nihmj/1273779621.full

On the other hand, the unit tangent bundle of the Klein bottle is this 3 manifold double covered by the torus

Unit (co)tangent bundle of Klein bottle

Is the unit tangent bundle always orientable?

• Hint: Start here. Commented Jan 29, 2022 at 16:45
• Once you got orientability of $TM$, construct a nonvanishing vector field on $TM$ normal to $UTM$. Commented Jan 29, 2022 at 16:56
• More generally, the sphere bundle of $E \to M$ is orientable if $w_1(E) = w_1(TM)$. If $\operatorname{rank} E > 1$, the converse is also true. Commented Jan 29, 2022 at 17:41

1. The total space $$TM$$ of tangent bundle to any manifold $$M$$ (orientable or not) is orientable, see here.
2. The total space $$UTM$$ of the unit tangent bundle is a hypersurface in $$TM$$ which admits a nowehere vanishing normal vector field (at each point $$(x,v)\in UTM$$ take the normal vector $$((x,v),v)$$).
3. Thus, $$UTM$$ is a cooriented hypersurface in an orientable manifold, hence, is itself orientable.