Quadratic Forms and Closed Geodesics

Indefinite Quadratic Forms
◮ Q(x, y) = ax 2 + bxy + cy 2 , d = b 2 − 4ac > 0
( ) t ( ) ( )
x x a b/2
◮ Q(x, y) = M M =
y y
b/2 c
◮ Which integers are represented by Q, i.e.
Q(x, y) = N,
x, y, N ∈ N
Quadratic Forms
and Closed
Geodesics
Y. Petridis
Quadratic Forms
Hyperbolic
surfaces
Closed
Geodesics
Spectral Theory
Equivalence of Quadratic Forms
Q ∼ Q ′ ⇒ they represent the same integers.
( ) 0 −1
◮ x 2 − 2y 2 ∼ −2x 2 + y 2 ◮ S =
1 0
◮ x 2 − 2y 2 ∼ x 2 + 2xy − y 2 ( ) 1 1
T =
0 1
◮ (x + y) 2 − 2y 2 = x 2 + 2xy − y 2
Q(x + y, y) = Q ′ (x, y) and Q ′ (x − y, y) = Q(x, y)