What is manifold? The following two pictures speak a great deal about the concept.

The official definition is: A simplicial k-complex is manifold if the link of every vertex looks like* a (k-1)-dimensional sphere.

To judge if it is a manifold triangle mesh, we see if every edge is contained in exactly two triangles, or just one along the boundary; if every vertex is contained in a single loop of triangles or a single fan along the boundary.

A data structure like manifold mesh capturing the connectivity is called topological data structure. It has profound usage. There are various types of topological data structure. One is Adjacency List.

Another form is incidence matrix, and signed incidence matrix.

This is easy to navigate but contains huge data points most of which are useless, hence sparse matrix is invented to store data thriftily. It’s effective but not easy to perform matrix operations/multiplications.

And what use frequently is half edge mesh. half edge to denote an edge with two opposite orientations. So then the next halfedge is deduced so is the face, so is the twin side.

We can give an verbose algebraic definition too, put into diagram