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Quadratic forms (i.e. homogeneous polynomials of degree two) are fundamental mathematical objects. For the ancient Greeks, quadratic forms manifested in the geometry of conic sections, and in Pythagoras’ theorem. Riemann recognized the importance of studying abstract smooth manifolds equipped with a field of infinitesimal quadratic forms (i.e. a *Riemannian metric*), giving rise to the theory of Riemannian manifolds. In contrast to more general norms, an inner product on a vector space enjoys a big group of symmetries; thus infinitesimal Riemannian geometry inherits all the richness of the representation theory of orthogonal groups, which organizes the various curvature tensors and Weitzenbock formulae. It is natural that quadratic forms should come up in so many distinct ways in differential geometry: one uses calculus to approximate a smooth object near some point by a linear object, and the “difference” is a second-order term, which can often be interpreted as a quadratic form. For example:

- If is a Riemannian manifold, at any point one can choose an orthonormal frame for , and exponentiate to obtain geodesic normal co-ordinates. In such local co-ordinates, the metric tensor satisfies and . The second order derivatives can be expressed in terms of the Riemann curvature tensor at .
- If is an immersed submanifold of Euclidean space, at every point there is a unique linear subspace that is tangent to at . The second order difference between these two spaces is measured by the
*second fundamental form*of , a quadratic form (with coefficients in the normal bundle) whose eigenvectors are the directions of (extrinsic)*principal curvature.*If has codimension one, the second fundamental form is easily described in terms of the*Gauss map*taking each point on to the unique unit normal to at that point, and using the flatness of the ambient Euclidean space to identify the normal spheres at different points with “the” standard sphere. The second fundamental form is then defined by the formula . For higher codimension, one considers Gauss maps with values in an appropriate Grassmannian. - If is a smooth function on a manifold , a
*critical point*of is a point at which (i.e. at which all the partial derivatives of in some local coordinates vanish). At such a point, one defines the*Hessian*, which is a quadratic form on , determined by the second partial derivatives of at such a point. If is a Levi-Civita connection on (determined by an Riemannian metric on compatible with the smooth structure) then . The condition that the Levi-Civita connection is torsion-free translates into the fact that the antisymmetric part of is equal to for any -form ; in this context, this means that the antisymmetric part of the Hessian vanishes — i.e. that it is symmetric (and therefore a quadratic form). If is a different connection, then for some -form , and therefore their values at agree, and is well-defined, independent of a choice of metric.

By contrast, *cubic forms* are less often encountered, either in geometry or in other parts of mathematics; their appearance is often indicative of unusual richness. For example: Lie groups arise as the subgroups of automorphisms of vector spaces preserving certain structure. Orthogonal and symplectic groups are those that preserve certain (symmetric or alternating) quadratic forms. The exceptional Lie group is the group of automorphisms of that preserves a generic (i.e. nondegenerate) alternating -form. One expects to encounter cubic forms most often in flavors of geometry in which the local transformation pseudogroups are bigger than the orthogonal group.

One example is that of -dimensional complex projective geometry. If is a domain in the Riemann sphere, one can think of as a geometric space in at least two natural ways: by considering the local pseudogroup of all holomorphic self-maps between open subsets of the Riemann sphere, restricted to (i.e. all holomorphic functions), or by considering only those holomorphic maps that extend to the entire Riemann sphere (i.e. the projective transformations: ). The difference between these two geometric structures is measured by a third-order term, called the *Schwarzian derivative*. If is homeomorphic to a disk, then we can think of as the image of the round unit disk under a uniformizing map . At every point there is a unique projective transformation that osculates to to second order at (i.e. has the same value, first derivative, and second derivative as at the point ); the (scaled) third derivative is the Schwarzian of at . In local co-ordinates, . Actually, although the Schwarzian is sensitive to third-order information, it should really be thought of as a quadratic form on the (one-dimensional) complex tangent space to .

Real projective geometry gives rise to similar invariants. Consider an immersed curve in the (real projective) plane. At every point, there is a unique osculating conic, that agrees with the immersed curve to second order. The *projective curvature* (really a cubic form) measures the third order deviation between these two immersed submanifolds at this point. See e.g. the book by Ovsienko and Tabachnikov for more details.

Another example is the so-called *symplectic curvature*. Let be a flat symplectic space; this could be ordinary Euclidean space with its standard symplectic form, or a quotient of such a space by a discrete group of translations. A linear subspace of through the origin is a *Lagrangian* subspace if it has (maximal) dimension , and the restriction of the symplectic form to is identically zero. A smooth submanifold of dimension is Lagrangian if its tangent space at every point is a Lagrangian submanifold. A Lagrangian submanifold of a flat symplectic space inherits a natural cubic form on the tangent space at every point, which can be defined in any of the following equivalent ways:

- If is a symplectic manifold and is a Lagrangian submanifold, then near any point one can find a neighborhood and choose symplectic coordinates so that is symplectomorphic to a neighborhood of some point in . Moreover, every other Lagrangian submanifold sufficiently close (in ) to can be taken in some possibly smaller neighborhood to be of the form , where is a smooth function on (well-defined up to a constant), thought of as a section of . In the context above, choose local symplectic coordinates (by a linear symplectic transformation) for which the flat space looks locally like and looks locally like . The condition that and are tangent at the origin means that the -jet of vanishes. The first nonvanishing term are the third partial derivatives of , which can be thought of as the coefficients of a (symmetric) cubic form on .
- If we choose a Euclidean metric on compatible with the flat symplectic structure, the second fundamental form of at some point is a quadratic form on with coefficients in the normal bundle to . The symplectic form identifies the normal to with the dual , so by contracting indices, one obtains a cubic form on . This form does not depend on the choice of Euclidean metric, since a different metric skews the normal bundle replacing it with . But since is Lagrangian, the identification of this normal bundle with is insensitive to the skewed term, and therefore independent of the choices.
- The space of all Lagrangian subspaces of is a symmetric space, homeomorphic to , sometimes called the
*Shilov boundary*of the Siegel upper half-space. If and is a tangent vector to in , then one obtains a symmetric quadratic form on in the following way. If is a transverse Lagrangian to , and is a -parameter family of Lagrangians starting at , then for small the Lagrangians and are transverse, and span . For any there is a unique decomposition . Define . Then is a symmetric bilinear form that vanishes on , and therefore descends to a form on that depends only on . A Lagrangian submanifold maps to by the Gauss map . One obtains a cubic form on associated to as follows: if then is a tangent vector to in , and therefore determines a quadratic form on ; this form is then evaluated on the vectors .

One application of symplectic curvature is to homological mirror symmetry, where the symplectic curvature associated to a Lagrangian family of Calabi-Yau -folds in determines the so-called “Yukawa 3-differential”, whose expression in a certain local coordinate gives the generating function for the number of rational curves of degree in a generic quintic hypersurface in . This geometric picture is described explicitly in the work of Givental (e.g. here). In another more recent paper, Givental shows how the topological recursion relations, the string equation and the dilaton equation in Gromov-Witten theory can be reformulated in terms of the geometry of a certain Lagrangian cone in a formal loop space (the geometric property of this cone is that it is *overruled* — i.e. each tangent space is tangent to the cone exactly along , where is a formal variable). This geometric condition translates into properties of the symplectic curvature of the Lagrangian cone, from which one can read off the “gravitational descendents” in the theory (let me add that this subject is quite far from my area of expertise, and that I come to this material as an interested outsider).

Cubic forms occur naturally in other “special” geometric contexts, e.g. holomorphic symplectic geometry (Rozansky-Witten invariants), affine differential geometry (related to the discussion of the Schwarzian above), etc. Each of these contexts is the start of a long story, which is best kept for another post.

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