  The intriguing question, why the present scale of the universe is free from
any perceptible footprints of rank-2 antisymmetric tensor fields? (generally
known as Kalb-Ramond fields) is addressed. A quite natural explanation of this
issue is given from the angle of higher-curvature gravity, both in four- and in
five-dimensional spacetime. The results here obtained reveal that the amplitude
of the Kalb-Ramond field may be actually large and play a significant role
during the early universe, while the presence of higher-order gravity
suppresses this field during the cosmological evolution, so that it eventually
becomes negligible in the current universe. Besides the suppression of the
Kalb-Ramond field, the extra degree of freedom in $F(R)$ gravity, usually known
as scalaron, also turns out to be responsible for inflation. Such F(R) gravity
with Kalb-Ramond fields may govern the early universe to undergo an
inflationary stage at early times (with the subsequent graceful exit) for wider
range of F(R) gravity than without antisymmetric fields.. Furthermore, the
models---in four- and five-dimensional spacetimes---are linked to observational
constraints, with the conclusion that the corresponding values of the spectral
index and tensor-to-scalar ratio closely match the values provided by the
Planck survey 2018 data.
