  We propose a geometric {approach to Lieb-Schultz-Mattis theorem for} quantum
many-body systems with discrete spin-rotation symmetries and lattice inversion
or rotation symmetry, but without translation symmetry assumed. Under
symmetry-twisting on a $(d-1)$-dimensional plane, we find that any
$d$-dimensional inversion-symmetric spin system possesses a doubly degenerate
spectrum when it hosts a half-integer spin at the inversion-symmetric point. We
also show that any rotation-symmetric generalized spin model with a projective
representation at the rotation center has a similar degeneracy under
symmetry-twisting. We argue that these degeneracies imply that {a unique
symmetric gapped ground state that is smoothly connected to product states} is
forbidden in the original untwisted systems -- generalized
inversional/rotational Lieb-Schultz-Mattis theorems without lattice translation
symmetry imposed. The traditional Lieb-Schultz-Mattis theorems with
translations also fit in the proposed framework.
