Given a Riemannian metric $g$ on a 2 dimensional manifold $M$, for $p\in M$ consider the quadratic surface (curve) $E$ around $0_p\in T_pM$ with coefficients equal to the fundamental tensor $g_{ij}$. That is $E=\{(x_1,x_2)\in R^2| \sum_{ij}g_{ij}x_ix_j=1\}$.
Is $E$ a rotated ellipse?
According to the definition of ellipse as quadric and the fact for the fundamental tensor that the determinant of [g_{ij}] is positive definite, the answer is yes. Indeed one has $$g_{11}x_1^2+g_{12}x_1x_2+g_{22}x_2^2=1$$ in which $g_{11}g_{22}-\frac{g_{12}^2}{4}>0$ that means we have an ellipse.