On the Ddvv Conjecture and the Comass in Calibrated Geometry (ii)
نویسنده
چکیده
where {e1, · · · , en} (resp. {ξ1, · · · , ξm}) is an orthonormal basis of the tangent (resp. normal space) at the point x ∈ M , and R,R are the curvature tensors for the tangent and normal bundles, respectively. In the study of submanifold theory, De Smet, Dillen, Verstraelen, and Vrancken [5] made the following DDVV Conjecture: Conjecture 1. Let h be the second fundamental form, and let H = 1 n trace h be the mean curvature tensor. Then ρ+ ρ ≤ |H| + c.
منابع مشابه
On the DDVV conjecture and the comass in calibrated geometry (I)
In this paper, we proved the first non-trivial case of the DDVV conjecture. Namely, for all 3 × 3 matrices, the DDVV inequality is valid. We also classified all the minimal submanifolds for which the equality holds.
متن کاملRecent Developments of the Ddvv Conjecture
Contents 1. Introduction 1 2. Relation to the comass problem in calibrated geometry 3 3. Relation to a conjecture of Böttcher and Wenzel 4 4. Relation to the Pinching theorems 6 5. A warming up exercise 9 6. Invariance 10 7. Sketch of the proofs 11 References 13
متن کاملCayley Form
Following an idea of Dadok, Harvey and Morgan, we apply the triality property of Spin(8) to calculate the comass of selfdual 4-forms on R. In particular, we prove that the Cayley form has comass 1 and that any self-dual 4-form realizing the maximal Wirtinger ratio (equation (1.5)) is SO(8)-conjugate to the Cayley form. We also use triality to prove that the stabilizer in SO(8) of the Cayley for...
متن کاملConvex Geometry of Orbits
We study metric properties of convex bodies B and their polars B, where B is the convex hull of an orbit under the action of a compact group G. Examples include the Traveling Salesman Polytope in polyhedral combinatorics (G = Sn, the symmetric group), the set of nonnegative polynomials in real algebraic geometry (G = SO(n), the special orthogonal group), and the convex hull of the Grassmannian ...
متن کاملیک مدل نیمه تجربی بهمنظور تخمین ابعاد جبهه رطوبتی در آبیاری قطرهای، تحت منبع نقطهای
For an appropriate drip irrigation system design, a prediction of soil wetting pattern is needed for a given soil texture. The wetting pattern geometry is a key factor for emitter distance determination as well as crop type. The geometry of the wetting bulb is dependent on the parameters such as soil hydraulic properties, emitter discharge and the irrigation time. This study has been conducted ...
متن کامل