Quadratic Forms and Closed Geodesics
Quadratic Forms and Closed Geodesics
Quadratic Forms and Closed Geodesics
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
Indefinite <strong>Quadratic</strong> <strong>Forms</strong><br />
◮ Q(x, y) = ax 2 + bxy + cy 2 , d = b 2 − 4ac > 0<br />
( ) t ( ) ( )<br />
x x a b/2<br />
◮ Q(x, y) = M M =<br />
y y<br />
b/2 c<br />
◮ Which integers are represented by Q, i.e.<br />
Q(x, y) = N,<br />
x, y, N ∈ N<br />
<strong>Quadratic</strong> <strong>Forms</strong><br />
<strong>and</strong> <strong>Closed</strong><br />
<strong>Geodesics</strong><br />
Y. Petridis<br />
<strong>Quadratic</strong> <strong>Forms</strong><br />
Hyperbolic<br />
surfaces<br />
<strong>Closed</strong><br />
<strong>Geodesics</strong><br />
Spectral Theory<br />
Equivalence of <strong>Quadratic</strong> <strong>Forms</strong><br />
Q ∼ Q ′ ⇒ they represent the same integers.<br />
( ) 0 −1<br />
◮ x 2 − 2y 2 ∼ −2x 2 + y 2 ◮ S =<br />
1 0<br />
◮ x 2 − 2y 2 ∼ x 2 + 2xy − y 2 ( ) 1 1<br />
T =<br />
0 1<br />
◮ (x + y) 2 − 2y 2 = x 2 + 2xy − y 2<br />
Q(x + y, y) = Q ′ (x, y) <strong>and</strong> Q ′ (x − y, y) = Q(x, y)