An elementary proof that rationally isometric quadratic forms are isometric
نویسندگان
چکیده
منابع مشابه
Linearization of Isometric Embeddings between Banach Spaces : an Elementary Approach
Mazur-Ulam’s classical theorem states that any isometric onto map between normed spaces is linear. This result has been generalized by T. Figiel [F] who showed that ifΦ is an isometric embedding from a Banach space X to a Banach space Y such that Φ(0) = 0 and vect [φ(X )] = Y , there exists a linear quotient map Q such that ‖Q‖ = 1 and Q ◦Φ= I dX . The third chapter of this short story is [GK] ...
متن کاملGunther’s Proof of Nash’s Isometric Embedding Theorem
Around 1987 a German mathematician named Matthias Gunther found a new way of obtaining the existence of isometric embeddings of a Riemannian manifold. His proof appeared in [1, 2]. His approach avoids the so-called Nash-Moser iteration scheme and, therefore, the need to prove smooth tame or Moser-type estimates for the inverse of the linearized operator. This simplifies the proof of Nash’s isom...
متن کاملOrbifold Lens Spaces That Are Isospectral but Not Isometric
We answer Mark Kac’s famous question [K], “can one hear the shape of a drum?” in the negative for orbifolds that are spherical space forms. This is done by extending the techniques developed by A. Ikeda on Lens Spaces to the orbifold setting. Several results are proved to show that with certain restrictions on the dimensionalities of orbifold Lens spaces we can obtain infinitely many pairs of i...
متن کاملOn Isometric and Minimal Isometric Embeddings
In this paper we study critial isometric and minimal isometric embeddings of classes of Riemannian metrics which we call quasi-κ-curved metrics. Quasi-κ-curved metrics generalize the metrics of space forms. We construct explicit examples and prove results about existence and rigidity. Introduction Definition: Let (M, g̃) be a Riemannian manifold. We will say g̃ is a quasi-κcurved metric if there ...
متن کاملRationally trivial quadratic spaces are locally trivial:III
It is proved the following. Let R be a regular semi-local domain containing a field such that all the residue fields are infinite. Let K be the fraction field of R. If a quadratic space (R, q : R → R) over R is isotropic over K, then there is a unimodular vector v ∈ R such that q(v) = 0. If char(R) = 2, then in the case of even n we assume that q is a non-singular space in the sense of [Kn] and...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Archiv der Mathematik
سال: 2014
ISSN: 0003-889X,1420-8938
DOI: 10.1007/s00013-014-0676-7