DDG 07 GENERALIZED CURVATURES J.M. Morvan Institut Camille Jordan - PDF document

DDG 07 GENERALIZED CURVATURES J.M. Morvan Institut Camille Jordan Universit� Claude Bernard Lyon 1, with ... V. Borrelli, D. Cohen-Steiner, B. Thibert. 1

... An Introduction ... "In what sense do two sets have to be close to each other, in order to guarantee that their curvature measures are close to each other ?" J. Milnor , (1994) ... Enigmatic question ... • What are the curvature measures of a set ? • What does mean : close to each other ? 2

... TWO QUESTIONS ... • Does a geometric property de�ned on the class of smooth objects have �an analo- gous� on the class of discrete objects? • De�ne a frame in which one can compare geometric properties of a smooth object and the corresponding geometric proper- ties of a discrete object close to it . 3

... EXAMPLE ... • On a smooth curve, one can de�ne its length l , its curvature k , its torsion t ... • On a smooth surface, one can de�ne its area A , its Gauss curvature G , its mean curvature H , its second fundamental form h , the lines of curvatures ... Can one de�ne these invariants on discrete objects like polyhedron for instance ? The main problem is to �nd GOOD DEFINITIONS 4

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HISTORY 1. The convex case, Steiner, 1840 2. The smooth case, Weyl, 1939 3. The introduction of geometric measure, Federer, 1958 4. The introduction of cohomology, Wint- gen, 1978 5. The recent works, (Zahle, Fu, etc... 1980 − 2007 ) 6

THE CURVATURES OF A SMOOTH HYPERSURFACE M : (oriented) smooth hypersurface of the (oriented) Euclidean space E N , < . >, ˜ ∇ . • Let ξ be the unit normal vector �eld to M . • The Weingarten tensor (shape operator) A de�ned on M : (1) A ( X ) = − ˜ ∇ X ξ. • The second fundamental form h of M : (2) < A ( X ) , Y > = h ( X, Y ) . • The principal curvatures ( λ 1 , ..., λ i , ..., λ ( N − 1) ) of M at m , and the principal vector �elds , 7

De�nition Let M be a smooth hypersurface of E N . For every , k = 0 , ..., ( N − 1) , the k − th - elementary symmetric function of the princi- pal curvatures of M Ξ k = { λ i 1 , ..., λ i k } is called the k − th -mean curvature of M . Remark : One has k = N − 1 det ( I + tA ) = Ξ k t k , (3) � k =0 • Ξ 0 = 1 ; • Ξ 1 is the trace of h , ( H = 1 n X 1 is called the mean curvature of M ). • Ξ N − 1 is the determinant of h , (usually denoted by G ), called the Gauss curva- ture of M .

CURVATURE MEASURES OF A SMOOTH HYPERSURFACE Let U be a domain of M . � ϕ k ( U ) = U Ξ k dv M is called the k -th Lipschitz-Killing curva- ture of M . • ϕ 0 ( U ) is the classical ( N − 1) -volume of U ; • ϕ N ( U ) is the total Gauss curvature of U . 8

Go back to the problem : How to de�ne these curvature measures for discrete subsets of E N ? A IMPORTANT TOOL : THE VOLUME By computing the volume of some tubular neighborhoods of a subset X , we can exhibit some geometric invariants of X ... 9

THE CONVEX CASE • K convex body, • K ǫ = { m ∈ E N , d ( m, K ) ≤ ǫ } , ε A • ∂K : convex hypersurface, • ∂K ǫ : parallel hypersurface at distance ǫ . 10

THE CONVEX CASE... STEINER FORMULA. Theorem - Let K be a convex body. Then, N Φ k ( K ) ǫ k , ∀ ǫ ≥ 0 . � V ( K ǫ ) = k =0 AN ADDITIVITY PROPERTY Theorem: If K 1 , K 2 an K 1 ∪ K 2 are convex, then ∀ k , Φ k ( K 1 ∪ K 2 ) = Φ k ( K 1 ) + Φ k ( K 2 ) − Φ k ( K 1 ∩ K 2 ) . CONTINUITY OF THE Φ k IN THE CONVEX WORLD Theorem: If a sequence of K n tends to K in the Hausdor� sense, then ∀ k , n →∞ Φ k ( K n ) = Φ k ( K ) . lim 12

Question: Compute explicitly the Φ ′ k s ? We will test on • smooth objects, • polyhedra. 13

THE CONVEX CASE... STEINER FORMULA FOR SMOOTH BODIES Steiner formula can be stated as follows : k = N Φ k ( K ) ǫ k , � Vol N ( K ǫ ) = k =0 with � Φ k ( K ) = C N ϕ k ( K ) = C N ∂ K Ξ k dv M . In particular, if K is a compact convex body in E 3 with smooth boundary, then Vol 3 ( K ǫ ) = ∂ K Hda ) ǫ 2 +1 � � ∂ K Gda ) ǫ 3 , Vol 3 ( K )+ A ( ∂ K ) ǫ +( 3( where H , (resp. G ), denotes the mean cur- vature, (resp. Gauss curvature) of ∂ K . 14

THE CONVEX CASE... STEINER FORMULA FOR POLYHEDRA We give the following De�nition 1 Let σ l be a l -dimensional face of a k -simplex σ k , ( l < k ) . Let q ∈ int( σ l ) . The following notions are independent of q . 1. The normal cone C ⊥ ( σ l , σ k ) to σ l : qx ∈ σ l ⊥ } . C ⊥ ( σ l , σ k ) = { x ∈ int( σ k ) : � 2. The basis of the normal cone C ⊥ ( σ l , σ k ) to σ l is L ( σ l , σ k ) = C ⊥ ( σ l , σ k ) ∩ S k − l − 1 , S k − l − 1 is the unit sphere centered where at q . 15

e q 3 σ � 3. The internal dihedral angle ( σ l , σ k ) is the measure of L ( σ l , σ k ) . The normalized internal dihedral angle ( σ l , σ k ) : � ( σ l , σ k ) ( σ l , σ k ) = . s k − l − 1 ( σ l , σ k ) ∗ is � 4. The external dihedral angle S k − l − 1 ob- the measure of the subset of tained by intersecting S k − l − 1 with the half- lines whose origin is q , and making an an- gle greater than π 2 with the interior of σ k .

t v The normalized external dihedral angle ( σ l , σ k ) ∗ : � ( σ l , σ k ) ∗ ( σ l , σ k ) ∗ = . s k − l − 1

THE CONVEX CASE... STEINER FORMULA FOR POLYHEDRA Theorem - Let K be a convex body of E N whose boundary ∂ K is a polyhedron. Then, k = N Φ k ( K ) ǫ k , � Vol N ( K ǫ ) = k =0 with Vol N − k ( σ N − k )( σ N − k , σ N ) ∗ . � Φ k ( K ) = σ N − k ⊂ σ N ⊂ K where • σ N − k denotes a generic ( N − k ) -face of ∂ K , • ( σ N − k , σ N ) ∗ denotes the normalized diedral external angle, 16

THE CONVEX CASE... STEINER FORMULA FOR POLYHEDRA IN E 3 ∠ ( a ) l ( a )) ǫ 2 + 4 Vol 3 ( K ǫ ) = Vol 3 ( K )+ A ( ∂ K ) ǫ +( � 3 πǫ 3 . a Remember the smooth case : � Hda ) ǫ 2 +1 � Vol 3 ( K ǫ ) = Vol 3 ( K )+ A ( ∂ K ) ǫ +( Gda ) ǫ 3 , 3( ∂ K ∂ K 17

Example : One de�ne the global mean curvature • of a smooth compact convexe surface S by � Φ 2 ( S ) = S Hda, • of a compact convex polyhedron P by: � Φ 2 ( P ) = ∠ ( a ) .l ( a ) . a Then 1. In both cases, Φ 2 ( A ∪ B ) = Φ 2 ( A )+Φ 2 ( B ) − Φ 2 ( A ∩ B ) . 2. Moreover, if P n tends to S in the Haus- dor� sense then � � lim ∠ ( a n ) .l ( a n ) = S Hda. n 18

HOWEVER, ALL THIS THEORY FAILS FOR NON CONVEX SUBSETS BECAUSE : • THE ADDITIVITY FORMULA IS NOT SATISFIED ; • THE CONTINUITY PROPERTY IS NOT SATISFIED. 19

... A CLASSICAL PROBLEM ... NON CONTINUITY OF THE AREA The situation is completely di�erent for sur- faces and their areas.... Let M be a C ∞ sur- face of E 3 . One can de�ne its area A ( M ) . It satis�es A ( M 1 ∪ M 2 ) = A ( M 1 )+ A ( M 2 ) −A ( M 1 ∩ M 2 ) . Let P be a polyhedron. One can de�ne its area which satis�es the same property. What about the continuity property? 20

THE LANTERN OF SCHWARZ 21

THE LANTERN OF SCHWARZ Let C be a cylinder of height l and radius R . Let P n,N be the lantern with N slices and 2 n triangles on each slice. The sequence P n,n 2 . The area of P n,n 2 sat- is�es: � R 2 π 4 + l 2 , A ( P n,n 2 ) → 2 πR 4 Then: A ( P n,n 2 ) → A ′ � = A ( C ) . However δ ( P n,n 2 , C ) → 0 (where δ is Hausdor� distance). 22

NON ADDITIVITY OF Vol K ǫ Vol ( I 1 ∪ I 2 ) ǫ � = Vol ( I 1 ) ǫ + Vol ( I 2 ) ǫ − Vol ( I 1 ∩ I 2 ) ǫ . p q r 23

... ANOTHER POINT OF VIEW ... THE NORMAL CYCLE Let M smooth hypersurface in E N . The good object to study is the Gauss map. G : M → E N × S N − 1 ⊂ E N × E N ≃ TE N m → G ( m ) = ( m, ξ ) CRUCIAL FACT : integrating particular di�erential forms of E N × S N − 1 on G ( U ) gives all the Lipschitz curvatures of U !! 24

INVARIANT ( N − 1) -FORMS ON E N × S N − 1 Let G the group of rigid motions of E N . G acts on E N × S N − 1 in a natural way. ω ∈ Λ N − 1 ( E N × S N − 1 ) is invariant (by G ) if φ ∗ ( ω ) = ω, for all φ ∈ G . One can build a basis ω 0 , ..., ω N of the space of invariant di�erential forms on E N × S N − 1 . 25

INVARIANT 2-FORMS ON E 3 × S 2 The vector space of invariant 2 -forms on E 3 × S 2 has dimension 4 . It is spent by ω A = y 1 dx 2 ∧ dx 3 + y 2 dx 3 ∧ dx 1 + y 3 dx 1 ∧ dx 2 . ω G = y 1 dy 2 ∧ dy 3 + y 2 dy 3 ∧ dy 1 + y 3 dy 1 ∧ dy 2 . ω H = y 1 ( dx 2 ∧ dy 3 + dy 2 ∧ dx 3 )+ y 2 ( dx 3 ∧ dy 1 + dy 3 ∧ dx 1 )+ y 3 ( dx 1 ∧ dy 2 + dy 1 ∧ dx 2 ) . 3 ) dx 1 ∧ dy 1 − y 1 y 2 dx 1 ∧ dy 2 − y 1 y 3 dx 1 ∧ dy 3 Ω = ( y 2 2 + y 2 − y 1 y 2 dx 2 ∧ dy 1 + ( y 2 3 ) dx 2 ∧ dy 2 n − y 2 y 3 dx 2 ∧ dy 3 1 + y 2 − y 1 y 3 dx 1 ∧ dy 3 − y 2 y 3 dx 3 ∧ dy 2 + ( y 2 3 ) dx 3 ∧ dy 3 . 2 + y 2 26

M smooth surface of E 3 � G ( U ) ω A = A ( U ); • • � G ( U ) ω G = � U G ; • � G ( U ) ω H = � U H ; � • G ( U ) Ω = 0 . 27

THE NORMAL CYCLE... (the case N=3) X → N ( X ) "Theorem" - To "each" compact sub- set X of E N , one can associate canonically an unique �interesting� recti�able Legendrian ( N − 1) -current N ( X ) of E N × S N − 1 . More- over, the map N satis�es the following addi- tivity formula : If A and B are two compact subsets of E 3 then, N ( A ∪ B ) = N ( A ) + N ( B ) − N ( A ∩ B ) . 28