View PDF - Project Euclid
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434 PHILLIP A. GRIFFITHS<br />
is the (pullback to (M) of the) first fundamental form of M, and that {oJ} is the<br />
connection matrix for the associated Riemannian connection.<br />
By the first relation in (1.3) we may think of (M) as an integral manifold of<br />
the differential system (<br />
to, O, &o. 0<br />
on (IR). By (1.2), the second of these equations is<br />
0 E Oa i ogaz<br />
which by the well-known Cartan lemma implies that<br />
(1.4) to. ’. h.oJo, h. h..<br />
The linear system of quadratic forms<br />
II h0.oo0 (R) e.<br />
is called the second fundamental form of M in N; for each unit normal<br />
e<br />
(II, ) h<br />
(dx, )<br />
R,y6toy<br />
By<br />
tr {} for M is given by<br />
ft dto oy A tore<br />
by (1.2)<br />
by (1.4). Setting<br />
1-<br />
. gives the usual second fundamental form of the projection of M into the +<br />
spanned by T( and the Caan structure equation, the curvature " ma-<br />
A to6, R,z6 -R6v<br />
it follows that the components of the Riemann curvature tensorR are given in<br />
terms of the 2 na fundamental form by
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