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In the round real projective plane, project the great-circle arc $(\cos t,\sin t,0)$ for $0\le t\le \pi$. Its endpoint and velocity match after antipodal identification, so it is a closed geodesic. On the sphere its tangent and the constant vector $(0,0,1)$ are parallel. Endpoint identification fixes the returned tangent and reverses the returned transverse vector. Its parallel transport is therefore a reflection. Nonorientability permits determinant-minus-one holonomy.

On vertices $1,\dots ,6$, take the simplicial complex generated by $123,124,135,146,156,236,245,256,345,346$. Every edge occurs in two facets. The vertex links, listed cyclically, are $23564$, $13654$, $12645$, $12536$, $13426$, and $14325$, respectively. Thus each vertex link is a cycle graph, and the realization is a compact connected surface without boundary. Its Euler characteristic is $6-15+10=1$, so the classification theorem for surfaces identifies it with the real projective plane. All fifteen possible edges occur, and the listed triangles are distinct three-element sets, so this is an honest abstract simplicial complex, not merely a polygon presentation.

Read a boundary word counterclockwise, giving every paired edge a label. A positive exponent means the edge arrow agrees with the boundary traversal, and an inverse means it disagrees. Identify equally labelled edges in their arrow directions. The sphere has a square word $ab{b}^{-1}{a}^{-1}$; the vertex count is three, giving Euler characteristic $3-2+1=2$. A closed orientable surface of genus $g\ge 1$ has wordA nonorientable surface which is the connected sum of surfaces of $k$ real projective planes has word $c_1{c}_1\cdots c_k{c}_k$. For $k=1$, this is a digon; the square word $abab$, identifying antipodal boundary points, is another model of the real projective plane. The Klein bottle has square word $ab{a}^{-1}b$, representing the nonorientable case $k=2$. The classification theorem for closed combinatorial surfaces says that these orientable and nonorientable families exhaust the connected closed cases.

In a triangle whose two sides at an apex are glued with that apex fixed, cross-sections become circles and the apex becomes their cone point. This quotient is a disc. A second triangle with boundary word $aa{b}^{-1}$ becomes a Möbius band after the $a$ sides are identified. Gluing the remaining $b$ boundary to the first disc caps the band and gives the real projective plane. The resulting two-face cell structure has two vertices, three edges and Euler characteristic one.