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An orientation of $S$ chooses a unit normal at every point. The Gauss map isFor the unit tangent $T=\dot{\alpha }$, the surface covariant derivative is the tangential projectionThe signed geodesic curvature isBecause $|T|=1$, $D_{s}T$ is perpendicular to $T$ and henceConsequently ${\kappa }_g=0$ means $D_{s}T=0$: the tangent is parallel along $\alpha$, so $\alpha$ is a geodesic.

Suppose that two closed geodesics ${\gamma }_1$ and ${\gamma }_2$ were disjoint. Under the usual convention that a closed geodesic curve is simple, the Jordan curve theorem on the sphere shows that they bound an annulus $A\subset S$. Its Euler characteristic is zero. Give $A$ the induced orientation and apply the Gauss-Bonnet theorem:Both boundary components are geodesics, so their geodesic curvature vanishes. It follows thatThis is impossible because $K>0$ everywhere and $A$ has positive area. Hence the two closed geodesics must intersect, as summarized by the intersection of closed geodesics on a positively curved sphere.

For unit speed on an oriented surface,These coefficients are normal and geodesic curvature; the curve is geodesic iff ${\kappa }_g=0$.

If a flat surface disc is replaced by a smooth disc agreeing with the original surface outside the same boundary, their boundary geodesic curvatures agree. Gauss-Bonnet theorem therefore forces the replacement’s total Gaussian curvature to be zero. If its curvature is everywhere nonnegative, continuity forces it to vanish everywhere. Thus a positive-curvature bulge cannot be glued this way without some negative curvature or a loss of smoothness.

A simple closed curve of length $L$ on the unit sphere enclosing the smaller area $A$ satisfiesEquality holds for a circle. A variational proof uses Gauss--Bonnet to turn area maximization at fixed length into geodesic-curvature minimization, proves that an extremizer has constant geodesic curvature and is planar, and evaluates its spherical cap.

If ambient isometries preserving a connected curve act transitively on it, then the magnitude of its geodesic curvature is constant. Continuity then makes the signed geodesic curvature either identically zero or of one fixed sign.