Isometric, holomorphic and symplectic reflections

Reflections in submanifolds

“…This should be contrasted with the result proved in [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15], characterizing the locally symmetric spaces by the property that the reflections in all geodesics are volume-preserving.…”

“…Then the reflection q~v in P is an isometry if and only if P is either a holomorphic totally geodesic submanifold or a totally real totally geodesic submanifold of dimension ½dim M. (See [7], and also [6] for further results. )…”

Locally symmetric immersions

“…ϕ m is said to be the local extrinsic symmetry at m for f. It is an involutive local diffeomorphism and U + m belongs to the fixed point set of ϕ m . Moreover, In [4] a criterion is derived for isometric reflections with respect to a submanifold. By using this we obtain the following criterion for isometric local extrinsic symmetries.…”

Symplectic reflections and complex space forms

“…For a Riemannian manifold one may also consider local reflections with respect to (embedded) submanifolds (see [1], [6], [7]). It is proved in [9] that (M, g) is locally symmetric if and only if the local reflection with respect to any geodesic is volumepreserving.…”

“…In [1] we considered mainly reflections with respect to submanifolds in Hermitian symmetric spaces. In particular, we proved…”