  Flux ropes (FRs) ejected from the Sun may change their geometrical
orientation during their evolution which directly affects their
geoeffectiveness. Therefore, it is crucial to understand how solar FRs evolve
in the heliosphere to improve our space weather forecasting tools. We analyze
15 coronal mass ejections (CMEs), with clear FR signatures, observed during the
decay of Solar Cycle 23 and rise of Solar Cycle 24. We estimate initial
orientations of the FRs at the origin using extreme ultraviolet observations of
post-eruption arcades and/or eruptive prominences. Then we reconstruct
multiviewpoint coronagraph observations of the CMEs from ~2 to 30 Rs with a
three-dimensional geometric representation of a FR to determine their
geometrical parameters. Finally, we propagate the FRs from ~30 Rs to 1 AU
through MHD-simulated background solar wind while using in-situ measurements at
1 AU of the associated magnetic cloud as a constraint for the propagation
technique. These methodology allows us to estimate the FR orientation all the
way from the Sun to 1 AU. We find that while the FRs deflection occurs
predominantly below 30 Rs, a significant amount of deflection and rotation
happens between 30 Rs and 1 AU. We compare the FR orientation to the local
orientation of the heliospheric current sheet (HCS). We find that slow FRs tend
to align with the streams of slow solar wind in the inner heliosphere. During
the solar cycle minimum the slow solar wind channel as well as the HCS usually
occupy the area in the vicinity of the solar equatorial plane, which in the
past led researchers to the hypothesis that FRs align with the HCS. Our results
show that exclusions from this rule are explained by interaction with the
Parker-spiralled background magnetic field, which dominates over the magnetic
interaction with the HCS in the inner heliosphere at least during solar minimum
conditions.
