Generalized Curvatures Part 9 pdf

170 17 Tubes Formula

Fig. 17.1 The map m →

m+tξ

ξ

M

m

m + tξ

Then, using the area formula (Theorem 19), we find that

VolN(Mn

ε ) = ∑

k

ε

t=0

ST ⊥Mn

Ξk(ξ )dvStT ⊥Mnt

k

dt

=

t=ε

t=0

SN−n−1

Mn ∑

k

Ξk(ξ )t

N−n−1+k

ω1 ∧...∧ωn ∧dt ∧dvSN−n−1

= (N −k)!

n! k!sk−1(

Mn

Ξk(ξ )dvM)εN−n+k

,

(17.16)

where (ωi

) denotes the dual frame of ei.

The left-hand side of (17.16) is a polynomial in ε, whose odd terms are null by

Proposition 13 since we assume here that Mn has no boundary. The even terms can

be written in terms of Rj by using the Gauss equation (17.9), from which we deduce

Theorem 51. The proof is similar for submanifolds with boundary.

17.2.2 Intrinsic Character of the Mk

Weyl made the following fundamental remark, which is an obvious consequence

of our previous computation: at each point of Mn, the Lipschitz–Killing curvature

forms depend only on the intrinsic geometry of Mn and are independent of the iso￾metric embedding. They can be computed with only the knowledge of the curvature

17.4 Partial Continuity of the Φk 171

tensor of Mn and the second fundamental form of the boundary ∂Mn, considered

as a hypersurface of Mn. This implies that two isometric immersions of the same

manifold Mn in EN have the same Lipschitz–Killing curvatures. In particular:

• The quantities Mk depend only on the intrinsic geometry of Mn and are conse￾quently independent of the embedding.

• Moreover, as for the coefficients Λk,N defined in the convex case (see (16.6),

Chap. 16), they are independent of the codimension. Once again, the Lipschitz–

Killing curvature forms and the global Lipschitz–Killing curvatures are intrinsic

invariants of Mn.

• When Mn has no boundary, the only non-null terms Mk are the even ones. These

coefficients are the same as those which appear in the Gauss–Bonnet formula,

generalized by Allendoerfer and Weil [3] and Chern [28] among others.

17.3 The Euler Characteristic

Recall that the Gauss–Bonnet theorem relates the Euler characteristic of a mani￾fold with its curvature tensor (see Chap. 10). If Mn is even dimensional and has no

boundary,

χ(Mn) = (−1)n/2

2nπn/2(n/2)!

Mn ∑

σ∈Sn

(−1)

σi

1...inΩi1

i2 ∧...∧Ωi(n−1)

in . (17.17)

With our notation, (17.14) implies that

ΦN(Mn) = MN(Mn) = sN−1

Mn

Rn = bNχ(Mn). (17.18)

If Mn has a boundary, we add the boundary term in Hn:

ΦN(Mn) = MN(Mn) = sN−1

Mn

Rn +

∂Mn

Hn = bNχ(Mn). (17.19)

Note the important fact that the previous equations show that the term ΦN has a

topological meaning, since the Euler characteristic is a topological invariant. This

result can be seen as a generalization of the convex case, since the Euler character￾istic of a convex subset is always equal to 1.

17.4 Partial Continuity of the Φk

It is clear that the continuity of the Φk mentioned for convex bodies is not satis￾fied for smooth submanifolds: for instance, using a construction analogous to the

Lantern of Schwartz (Sect. 3.1.3), it is easy to construct an example of sequence of