aspg-21-1788-1791475402 20261008160322 American Scientific Publishing Group admin@americaspg.com American Scientific Publishing Group International Journal of Neutrosophic Science IJNS 2692-6148 2690-6805 2023 21 1 Erick Gonzalez-Caballero Technological University of Havana, Havana, Cuba NeutroGeometries are those geometric structures where at least one definition, axiom, property, theorem, among others, is only partially satisfied. In AntiGeometries at least one of these concepts is never satisfied. Smarandache Geometry is a geometric structure where at least one axiom or theorem behaves differently in the same space, either partially true and partially false, or totally false but its negation done in many ways. This paper offers examples in images of nature, everyday objects, and celestial bodies where the existence of Smarandechean or NeutroGeometric structures in our universe is revealed. On the other hand, a practical study of surfaces with characteristics of NeutroGeometry is carried out, based on the properties or more specifically NeutroProperties of the famous quadrilaterals of Saccheri and Lambert on these surfaces. The article contributes to demonstrating the importance of building a theory such as NeutroGeometries or Smarandache Geometries because it would allow us to study geometric structures where the well-known Euclidean, Hyperbolic or Elliptic geometries are not enough to capture properties of elements that are part of the universe, but they have sense only within a NeutroGeometric framework. It also offers an axiomatic option to the Riemannian idea of Two-Dimensional Manifolds. In turn, we prove some properties of the NeutroGeometries and the materialization of the symmetric triad <Geometry>, <NeutroGeometry>, and <AntiGeometry>. 2023 14 32 1788 https://creativecommons.org/licenses/by/4.0/ 10.54216/IJNS.210102 https://www.americaspg.com/journal/21/article/1788 https://www.americaspg.com/storage/91683983004.pdf