History Of Mathematics Back cover copy This volume of selected academic papers demonstrates the significance of the contribution to mathematics made by Manfredo P. Twice a Guggenheim Fellow and the winner of many prestigious national and international awards, the professor at the institute of Pure and Applied Mathematics in Rio de Janeiro is well known as the author of influential textbooks such as Differential Geometry of Curves and Surfaces. Aspects covered in the featured papers include relations between curvature and topology, convexity and rigidity, minimal surfaces, and conformal immersions, among others. Offering more than just a retrospective focus, the volume deals with subjects of current interest to researchers, including a paper co-authored with Frank Warner on the convexity of hypersurfaces in space forms. It also presents the basic stability results for minimal surfaces in the Euclidean space obtained by the author and his collaborators.
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Niky Kamran N. April 3, 1 Introduction. To the best of our knowledge, the study of the local symplectic invariants of submanifolds of Euclidean space was initiated by Chern and Wang in , . They considered mainly the case of curves and hypersurfaces, and obtained structure equations defining a set of local symplectic differential invariants for these objects.
This is in contrast with the case in which one considers the full infinite-dimensional symplectomorphism group of the ambient R2n. Indeed, in the latter case, the theorem of Darboux implies that submanifolds have no local differential invariants. These lie beyond the scope of this work. The case we are interested in is much closer in spirit to ordinary Euclidean or affine 1 differential geometry, where the ambient space is a homogeneous space for the action of a finite-dimensional Lie group, and where submanifolds enjoy a wealth of local differential invariants.
Our purpose in this paper is to further develop the line of research initiated in  by constructing explicitly a complete set of local symplectic invariants for curves, through two different approaches. The structure functions of the frame are then the local invariants of curves. We will sometimes refer to these lo- cal differential invariants as the symplectic curvatures of the curve. See also [9,12] for further developments and applications of the equivariant approach to moving frames.
We show explicitly that these approaches lead to the same Frenet frame and to the same same set of local differential invariants. We also investigate the case of curves in R4 having the property that all their differential invariants are constant. By a general theorem of Cartan, , we know that this is the case if and only if the curve is the orbit of a one-parameter subgroup of the affine symplectic group. We give a classification of these curves according to the algebraic type of the spectrum of their corresponding Frenet matrices.
We would like to mention that there has been some recent work related to the symplectic geometry of submanifolds of Euclidean space including ,  and . However,  deals with the specific case of fanning curves in a symplec- tic manifold, while  constructs differential invariants for two other actions of the affine symplectic group, which are thus different from ours. Moreover, our approach yields not only the invariants, but also a Frenet frame, obtained 2 in analogy with the classical constructions from Euclidean geometry, and the method of moving frames.
This Frenet frame is indispensible when trying to re- construct the parametrized curves corresponding to a given choice of symplectic invariants.
Similar remarks apply to , where the authors consider the prob- lem of computing the differential invariants for different linear actions of the symplectic group, corresponding to different irreducible representations of the symplectic group. However, the explicit formulas for the differential invariants are only obtained for the lowest dimensional symplectic groups. Our paper is organized as follows. After reviewing in Section 2 some basic definitions from symplectic geometry, we introduce in Section 3 the concepts of symplectic arc length and symplectic regular curve, which takes care of the reparametrization freedom for the curves under consideration.
Section 4 is de- voted to the construction by successive differentiations of the symplectic Frenet frame and the corresponding local differential invariants.
We then prove an exis- tence and uniqueness theorem for curves with prescribed symplectic curvatures, analogous to the corresponding theorem in Euclidean geometry.
In Section 5, we show by explicit computation that the same Frenet frame and local differ- ential invariants can be constructed by the method of moving frames, using the algorithm described in . Section 7 is devoted to a general discussion of the case of curves in R4 of constant symplectic curvatures.
An algebraic classifica- tion of the spectrum of the Frenet matrix is given in terms of conditions on the numerical values of the constant symplectic curvatures. A rigid symplectic 1 By definition, a symplectic inner product is a nondegenerate, skew-symmetric bilinear form on the underlying vector space.
There is, obviously, no requirement of positive definite- ness. With no loss of generality, we may assume that the left-hand-side in 3. The latter is geometrically interpreted in 2 As defined, the symplectic arc length could be negative. Remark : We use dots to denote derivatives with respect to an arbitrary parametrization t, reserving primes to indicate derivatives with respect to sym- plectic arc length, which from here onwards we denote by s.
The following proposition shows that there is no loss of generality in as- suming that any symplectic regular curve can be re-parametrized by symplectic arclength. To any such curve, we associate an adapted symplectic frame a1 ,. This frame is defined recursively. Then the frame a1 ,. Proof: The proof proceeds by induction.
We will verify a few of the relations and leave the remaining ones to the reader as an exercise. We have, using 4. The proof of the remaining orthonormality conditions is similar, and will there- fore be omitted. We remark that it follows from the definitions 4. The first of the structure equations is a direct consequence of the definition 4. Then 4. Theorem 1 Let H2 ,. The existence of a Frenet frame satisfying the structure equations 4. It is now straightforward to show that any frame satisfying the orthonormality conditions 2.
Hence the solution of 4. These expressions are fairly simple when the symplectic regular curve is parametrized by symplectic arc length, but they become quite complicated in a more general parametrization. In this section, we review the method of moving coframes developed in  in a general setting. The following section will apply the method to the con- text at hand: symplectic curves. Throughout this section, G will denote an r-dimensional Lie group that acts smoothly on an m-dimensional manifold M.
We also assume, for the purposes of exposition, that G acts transitively on M , although this is inessential for the applicability of the moving frame method. Since we are dealing with local properties, there is no loss in generality in assuming that T is a subset of Euclidean space.
Again, the moving coframe method can equally well handle restricted reparametrization groups and pseudo-groups, but we will focus our attention to the simplest case here. These one- forms play the role of Maurer—Cartan forms for the reparametrization pseudo- group, . One then iterates the process until there are no remaining group parameters. The nonconstant coefficients of the resulting linear dependencies among the pulled-back Maurer—Cartan forms will give a complete system of differential invariants.
Two such curves are equivalent if the first can be mapped to the second under a rigid symplectic motion up to a reparametrization. We will show how to use the moving coframe method outlined in Section 5 in this particular context.
The initial step in the algorithm is to normalize the order zero lifted invariants 5. Next, we substitute the initial normalizations 6. Any resulting linear dependencies will give additional lifted differ- ential invariants.
Substituting 6. Remark : Most curves that fail to satisfy the nondegeneracy requirement 6. The only curves that do not admit any moving frame whatsoever are the totally singular curves, meaning those for which the group G fails to act locally freely on their jets of any order. See Section 7 for further results in this direction. Our normalization 6. Invariance of the reparametrization Maurer—Cartan form 6. Indeed, multiplying both sides of 6. The first two coefficients in 6.
Thus, as a consequence of the normalization 6. Let us implement the basic induction step to complete the moving frame computation. Owing to the order of normalization, the coefficients of e1 ,. For this choice of normalization constants, equation 6. Using 6. The coefficients of e1 ,. The remarkable fact is that most of the coefficients in the inductive normal- ization equations 6.
An easy induction on k, sinilar to that in the proof of Theorem 1, proves our main result. Corollary 3 The moving frame A, b given by 6. We have thus recovered the same moving frame, structure equations and a complete system of generating differential invariants as before. The four-dimensional case is the lowest-dimensional case of interest from a symplectic point of view since the case where the ambient space is R2 corresponds to the 26 unimodular affine geometry of plane curves, where it is well known that the curves with constant invariant are precisely the conics, .
One way to proceed would be to use the Theorem of Cartan which we referred to in the Introduction, which implies that the curves with constant symplectic curvatures are precisely the orbits of the one-parameter subgroups of the affine symplectic group in four variables, .
This would require the determination of all such one parameter subgroups. We shall proceed more directly by integrating the Frenet equations in the case in which all the symplectic curvatures are con- stant. This means of course that we restrict our attention to curves which admit a symplectic Frenet frame.
The eigenvalues of the Frenet matrix appearing in the right hand side of 7. Indeed, the p spectrum is restricted by virtue of the constraints 7.
Alvarez Paiva and C. Anderson, Introduction to the variational bicomplex, Contemp. Barbosa and B. A 37 , — Calabi, P. Olver and A. Tannenbaum, Affine geometry, curve flows, and invariant numerical approximations, Adv. Chern and H. Wang, Differential geometry in symplectic space, Sci- ence Report Nat.
Tsing Hua Univ.
Mathematical Sciences Research Institute
Membro do Com. Soliton solutions to the curve shortening flow on the sphere. The mean curvature flow by parallel hypersurfaces. Geometry Seminar, Istanbul University. On Backlund and Ribaucour transformations for surfaces with constant negative curvature. Geometry Seminar, Lehigh University.
Manfredo P. do Carmo - Selected Papers