The metric in the superspace of Riemannian metrics and its relation to gravity

H.-J. Schmidt

RECORD

The metric in the superspace of Riemannian metrics and its relation to gravity

H.-J. Schmidt

General Relativity and Quantum Cosmology · Submitted 1 Sep 2001

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Just one of this class has the property to bring the lagrangian of General Relativity into the form of a classical particle's motion.

how it is stated · asserted 1.00 read the record →

Comment (0) Read on arXiv ↗

Cite ▾

Claim paper

The compiled record

The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Open the compiled record

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,860 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.49 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.10 · compares with prior results 0.07 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

The space of all Riemannian metrics is infinite-dimensional. Nevertheless a great deal of usual Riemannian geometry can be carried over. The superspace of all Riemannian metrics shall be endowed with a class of Riemannian metrics; their curvature and invariance properties are discussed. Just one of this class has the property to bring the lagrangian of General Relativity into the form of a classical particle's motion. The signature of the superspace metric depends in a non-trivial manner on the signature of the original metric, we derive the corresponding formula. Our approach is a local one: the essence is a metric in the space of all symmetric rank-two tensors, and then the space becomes a warped product of the real line with an Einstein space.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The space of all Riemannian metrics is infinite-dimensional.	background 1.00	0.09	0.03	asserted 1.00
s1	Nevertheless a great deal of usual Riemannian geometry can be carried over.	background 0.96	0.11	0.06	asserted 0.98
s2	The superspace of all Riemannian metrics shall be endowed with a class of Riemannian metrics; their curvature and invariance properties are discussed.	contribution 0.96	0.62	0.15	asserted 0.84
s3	Just one of this class has the property to bring the lagrangian of General Relativity into the form of a classical particle's motion.	result 0.96	0.71	0.47	asserted 1.00
s4	The signature of the superspace metric depends in a non-trivial manner on the signature of the original metric, we derive the corresponding formula.	result 0.99	0.92	0.16	asserted 1.00
s5	Our approach is a local one: the essence is a metric in the space of all symmetric rank-two tensors, and then the space becomes a warped product of the real line with an Einstein space.	method 0.84	0.84	0.18	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper

Original abstract

The space of all Riemannian metrics is infinite-dimensional. Nevertheless a great deal of usual Riemannian geometry can be carried over. The superspace of all Riemannian metrics shall be endowed with a class of Riemannian metrics; their curvature and invariance properties are discussed. Just one of this class has the property to bring the lagrangian of General Relativity into the form of a classical particle's motion. The signature of the superspace metric depends in a non-trivial manner on the signature of the original metric, we derive the corresponding formula. Our approach is a local one: the essence is a metric in the space of all symmetric rank-two tensors, and then the space becomes a warped product of the real line with an Einstein space.

Human review

How Pith checks papers → · Full audit of this paper →

How the literature uses it

2 arXiv papers cite this in 2 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 1 sentence without a confident label

Cites it as background 1 paper

For example, in quantum field theory $E$ is a spinor bundle on $M$ (see [ref] ), in gauge field theory $E$ is a bundle of connections on a principal bundle over $M$ (see [ref] ), and in general relativity $E$ is the bundle of Riemannian metrics on $M$ (see [ref] ).

Euler-Lagrange equations for functionals defined on Fr\'echet manifolds · Introduction · 0.98

Discussion

Know something about this paper? Leave a comment, or highlight any passage above to comment on it.