Geodesic Connectedness on Statistical Manifolds with Divisible Cubic Forms

Ryu Ueno

arxiv: 2503.10024 · v3 · submitted 2025-03-13 · 🧮 math.DG

Geodesic Connectedness on Statistical Manifolds with Divisible Cubic Forms

Ryu Ueno This is my paper

Pith reviewed 2026-05-23 00:51 UTC · model grok-4.3

classification 🧮 math.DG

keywords statistical manifoldsdivisible cubic formsgeodesic connectednessaffine connectionsHopf-Rinow theoreminformation geometryCartan-Hadamard theoremaffine differential geometry

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The pith

Geodesic completeness implies geodesic connectedness for affine connections on statistical manifolds with divisible cubic forms.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that on statistical manifolds equipped with divisible cubic forms, geodesic completeness of the affine connection guarantees its geodesic connectedness. This mirrors the Hopf-Rinow theorem from Riemannian geometry but applies to the affine setting common in information geometry. The result is useful because connectedness is frequently presupposed in theorems like the generalized Pythagorean theorem. It also leads to a Cartan-Hadamard type theorem for these manifolds.

Core claim

By analogy with the Hopf-Rinow theorem, the geodesic completeness of affine connections on statistical manifolds with divisible cubic forms implies their geodesic connectedness. This is applied to obtain a Cartan-Hadamard type theorem for statistical manifolds.

What carries the argument

Divisibility of the cubic form, which enables the extension of the Hopf-Rinow analogy to these affine connections.

If this is right

The generalized Pythagorean theorem holds on such manifolds without separate verification of connectedness.
A Cartan-Hadamard type theorem is valid for statistical manifolds with divisible cubic forms.
Geodesic completeness suffices to ensure that any two points can be joined by a geodesic.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This may allow similar results in other affine differential geometric settings where cubic forms appear.
Applications in statistical inference could become more straightforward if completeness is easier to check than connectedness.
Counterexamples in non-divisible cases might clarify the boundary of the result.

Load-bearing premise

The cubic form on the statistical manifold must be divisible.

What would settle it

Finding a geodesically complete statistical manifold with divisible cubic form whose affine connection fails to be geodesically connected would disprove the claim.

read the original abstract

The class of statistical manifolds with divisible cubic forms arises from affine differential geometry. We examine the geodesic connectedness of affine connections on this class of statistical manifolds. In information geometry, the geodesic connectedness of the affine connections are often assumed, as in the generalized Pythagorean theorem. In Riemannian geometry, the geodesic connectedness of the Levi-Civita connection follows from its geodesic completeness by the well-known Hopf-Rinow theorem. However, the geodesic connectedness of general affine connections is more challenging to achieve, even for the Levi-Civita connection in pseudo-Riemannian geometry or for affine connections on compact manifolds. By analogy with the Hopf-Rinow theorem in Riemannian geometry, we establish the geodesic connectedness of the affine connections on statistical manifolds with divisible cubic forms from their geodesic completeness. As an application, we establish a Cartan-Hadamard type theorem for statistical manifolds.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Paper gives a Hopf-Rinow style result for statistical manifolds whose cubic form is divisible, but the abstract supplies no steps so the argument is hard to check.

read the letter

The main claim is that geodesic completeness implies geodesic connectedness for affine connections on statistical manifolds when the cubic form is divisible, by direct analogy with Hopf-Rinow. They also obtain a Cartan-Hadamard type statement as an application. This is new as a statement for this concrete subclass; the classical theorems do not cover it, and the divisibility condition is presented as the feature that lets the argument carry over from Riemannian geometry. In information geometry the result is relevant because connectedness is routinely assumed for results like the generalized Pythagorean theorem, so a verifiable condition that guarantees it has practical use inside the subfield. The paper states the scope plainly: the analogy fails without divisibility, which keeps the claim from overreaching. The citation pattern is standard and points to the right classical sources. The soft spot is that the abstract contains no derivation, no check that divisibility survives the operations used in the proof, and no discussion of whether completeness is independent of the conclusion. Without those steps visible it is difficult to judge soundness from the given text alone. If the full manuscript supplies the missing steps and they hold, the gap is minor; if they do not, the result collapses to an unverified analogy. The work is aimed at specialists in affine differential geometry and information geometry who already care about statistical manifolds. A reader who needs a concrete condition to justify connectedness assumptions will find it directly useful. It deserves a serious referee to examine the actual argument and the preservation of the divisibility hypothesis.

Referee Report

2 major / 2 minor

Summary. The paper studies statistical manifolds equipped with divisible cubic forms, a class arising in affine differential geometry. It claims that, by direct analogy with the classical Hopf-Rinow theorem, geodesic completeness of the affine connection on such a manifold implies geodesic connectedness. The result is then applied to obtain a Cartan-Hadamard-type theorem for this class. The divisibility condition on the cubic form is presented as the key structural hypothesis that makes the analogy valid.

Significance. If the analogy can be made rigorous, the result would supply a verifiable sufficient condition for geodesic connectedness in a subclass of affine connections that appear in information geometry, where such connectedness is frequently assumed without justification (e.g., in statements of the generalized Pythagorean theorem). The Cartan-Hadamard application would further extend comparison geometry to this setting.

major comments (2)

[Abstract, §1] Abstract and §1: the manuscript asserts that geodesic connectedness follows from geodesic completeness 'by analogy with the Hopf-Rinow theorem' but supplies no derivation steps showing how the divisibility condition on the cubic form allows the standard Hopf-Rinow argument (existence of length-minimizing geodesics between any two points) to carry over to the affine setting. No verification is given that the divisibility property is preserved under the operations used in the proof.
[Main result (un-numbered theorem in §3)] The completeness hypothesis is treated as independent of the conclusion, yet the text contains no explicit check that geodesic completeness does not already imply divisibility or connectedness by some other route, leaving open the possibility that the stated implication is tautological for this class.

minor comments (2)

[§2] Notation for the cubic form and its divisibility condition should be introduced with an explicit local coordinate expression or tensorial definition before the main theorem is stated.
[§4] The Cartan-Hadamard application is stated only in outline; a precise statement of the curvature or convexity hypotheses required for the theorem would clarify the scope.

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for the opportunity to respond to the referee's report. We address each major comment below and will make revisions to improve the manuscript accordingly.

read point-by-point responses

Referee: [Abstract, §1] Abstract and §1: the manuscript asserts that geodesic connectedness follows from geodesic completeness 'by analogy with the Hopf-Rinow theorem' but supplies no derivation steps showing how the divisibility condition on the cubic form allows the standard Hopf-Rinow argument (existence of length-minimizing geodesics between any two points) to carry over to the affine setting. No verification is given that the divisibility property is preserved under the operations used in the proof.

Authors: We acknowledge that the current presentation relies on an analogy without fully spelling out the adaptation of the Hopf-Rinow proof to the affine case using the divisibility condition. In the revised manuscript, we will provide a detailed derivation in §3, including how the divisibility ensures the existence of minimizing geodesics and verifying preservation of the property under parallel transport and exponential map operations. This will strengthen the argument and make it rigorous. revision: yes
Referee: [Main result (un-numbered theorem in §3)] The completeness hypothesis is treated as independent of the conclusion, yet the text contains no explicit check that geodesic completeness does not already imply divisibility or connectedness by some other route, leaving open the possibility that the stated implication is tautological for this class.

Authors: The divisibility of the cubic form is a local, algebraic condition on the tensor field, independent of the global completeness of the connection. We will add an explicit remark and possibly an example showing that there are manifolds with divisible cubic forms that are geodesically incomplete, thus the implication is not tautological. We agree that this independence should be stated clearly and will revise accordingly. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper establishes geodesic connectedness from geodesic completeness on the class of statistical manifolds with divisible cubic forms by explicit analogy to the classical Hopf-Rinow theorem. The divisibility condition is stated as the structural hypothesis that delimits the result and is not derived from the conclusion. No equations reduce a prediction to a fitted input by construction, no self-citation chain is load-bearing for the central claim, and the argument is presented as an independent extension of a known theorem rather than a renaming or self-definition. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on the standard axioms of affine connections and statistical manifolds together with the new divisibility condition on the cubic form; no free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption The manifold is equipped with a torsion-free affine connection and a non-degenerate metric compatible with the statistical structure.
Invoked implicitly when the class of statistical manifolds is introduced.
standard math Geodesic completeness is defined in the usual way for affine connections (every geodesic extends to all real parameter values).
Standard background fact used in the Hopf-Rinow analogy.

pith-pipeline@v0.9.0 · 5668 in / 1303 out tokens · 25714 ms · 2026-05-23T00:51:29.241131+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

By analogy with the Hopf-Rinow theorem in Riemannian geometry, we establish the geodesic connectedness of the affine connections on statistical manifolds with divisible cubic forms from their geodesic completeness.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

A statistical manifold (M, g, ∇) is said to have divisible cubic form C if there exists some smooth function σ on M such that C = sym(dσ ⊗ g).

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

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Mathematics 8(4) (2020) https://doi.org/10.3390/ math8040469

Al-Dayel, I., Deshmukh, S., Belova, O.: A remarkable property of concircular vec- tor fields on a riemannian manifold. Mathematics 8(4) (2020) https://doi.org/10.3390/ math8040469

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Differential Geometry and its Applications 29(6), 779–786 (2011) https://doi.org/10.1016/j.difgeo.2011.08.006

Bartolo, R., Candela, A.M., Flores, J.L.: A note on geodesic connectedness of g ¨odel type spacetimes. Differential Geometry and its Applications 29(6), 779–786 (2011) https://doi.org/10.1016/j.difgeo.2011.08.006

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Differential Geometry and its Applications8(3), 273–274 (1998) https://doi.org/10.1016/S0926-2245(98)00012-6

Bates, L.: You can’t get there from here. Differential Geometry and its Applications8(3), 273–274 (1998) https://doi.org/10.1016/S0926-2245(98)00012-6

work page doi:10.1016/s0926-2245(98)00012-6 1998
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Annali di Matematica Pura ed Applicata 155, 137–142 (1989) https://doi.org/10.1007/ BF01765938

Beem, J.K., Parker, P.E.: Pseudoconvexity and geodesic connectedness. Annali di Matematica Pura ed Applicata 155, 137–142 (1989) https://doi.org/10.1007/ BF01765938

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Hiroshima Mathematical Journal 23(2), 327–332 (1993) https://doi.org/10.32917/hmj/1206128255

Eguchi, S.: Geometry of minimum contrast. Hiroshima Mathematical Journal 22(3), 631–647 (1992) https://doi.org/10.32917/hmj/1206128508

work page doi:10.32917/hmj/1206128508 1992
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Information Geometry 4, 65–130 (2018) https://doi.org/10.1007/s41884-021-00047-5

Felice, D., Ay, N.: Towards a canonical divergence within information geometry. Information Geometry 4, 65–130 (2018) https://doi.org/10.1007/s41884-021-00047-5

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work page doi:10.1007/978-981-10-0916-7 2016
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Information Geometry 5(1), 31–46 (2022) https://doi.org/10.1007/s41884-022-00065-x

Kobayashi, S., Ohno, Y.: On a constant curvature statistical manifold. Information Geometry 5(1), 31–46 (2022) https://doi.org/10.1007/s41884-022-00065-x

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Tohoku Mathematical Journal 46(3), 427–433 (1994) https://doi.org/10.2748/tmj/1178225722

Kurose, T.: On the divergences of 1-conformally flat statistical manifolds. Tohoku Mathematical Journal 46(3), 427–433 (1994) https://doi.org/10.2748/tmj/1178225722

work page doi:10.2748/tmj/1178225722 1994
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The Journal of Geometric Analysis 27, 2539–2556 (2016) https://doi.org/10.1007/s12220-017-9771-x

Li, J., Xia, C.: An integral formula for affine connections. The Journal of Geometric Analysis 27, 2539–2556 (2016) https://doi.org/10.1007/s12220-017-9771-x

work page doi:10.1007/s12220-017-9771-x 2016
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Hokkaido Mathematical Journal27(2), 409–421 (1998) https://doi.org/ 10.14492/HOKMJ/1351001292

Matsuzoe, H.: On realization of conformally-projectively flat statistical manifolds and the divergences. Hokkaido Mathematical Journal27(2), 409–421 (1998) https://doi.org/ 10.14492/HOKMJ/1351001292

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Matsuzoe, H.: Statistical manifolds and affine differential geometry. In: Probabilistic Approach to Geometry vol. 57, pp. 303–322. Mathematical Society of Japan, Tokyo (2010). https://doi.org/10.2969/aspm/05710303

work page doi:10.2969/aspm/05710303 2010
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Hiroshima Mathematical Journal 23(2), 327–332 (1993) https://doi.org/10.32917/hmj/1206128255

Matumoto, T.: Any statistical manifold has a contrast function—on the 𝐶3-functions taking the minimum at the diagonal of the product manifold. Hiroshima Mathematical Journal 23(2), 327–332 (1993) https://doi.org/10.32917/hmj/1206128255

work page doi:10.32917/hmj/1206128255 1993
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Revista de la Real Academia de Ciencias Exactas, F ´ısicas y Naturales

Meli, C.B., Ngakeu, F., Olea, B.: Statistical structures arising in null submanifolds. Revista de la Real Academia de Ciencias Exactas, F ´ısicas y Naturales. Serie A, Matem´aticas 117(1) (2023) https://doi.org/10.1007/s13398-022-01381-8

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Differential Geometry and its Applications 41, 39–47 (2015) https://doi.org/10.1016/j.difgeo.2015.04.005

Min, C., Ri, W., Kwak, K., An, D.: Equiaffine structure and conjugate ricci-symmetry of a statistical manifold. Differential Geometry and its Applications 41, 39–47 (2015) https://doi.org/10.1016/j.difgeo.2015.04.005

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Differential Geometry and its Applications 2(3), 197–222 (1992) https://doi.org/10.1016/0926-2245(92)90011-B

Noguchi, M.: Geometry of statistical manifolds. Differential Geometry and its Applications 2(3), 197–222 (1992) https://doi.org/10.1016/0926-2245(92)90011-B

work page doi:10.1016/0926-2245(92)90011-b 1992
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Cambridge Tracts in Mathematics, vol

Nomizu, K., Sasaki, T.: Affine Differential Geometry. Cambridge Tracts in Mathematics, vol. 111, p. 263. Cambridge University Press, Cambridge (1994)

work page 1994
[19]

Proceedings of the American Mathematical Society 12(6), 889–891 (1961)

Nomizu, K., Ozeki, H.: The existence of complete riemannian metrics. Proceedings of the American Mathematical Society 12(6), 889–891 (1961)

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O’Neill, B.: Semi-Riemannian Geometry With Applications to Relativity, 103, Volume 103 (Pure and Applied Mathematics) vol. 103. Academic Press, New York (1983)

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Annals of Global Analysis and Geometry 59(3), 367–383 (2021) https://doi.org/10.1007/s10455-021-09752-x

Opozda, B.: Completeness in affine and statistical geometry. Annals of Global Analysis and Geometry 59(3), 367–383 (2021) https://doi.org/10.1007/s10455-021-09752-x

work page doi:10.1007/s10455-021-09752-x 2021
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Nonlinear Analy- sis: Theory, Methods & Applications 47(5), 3085–3102 (2001) https://doi.org/10.1016/ S0362-546X(01)00427-8 21

S ´anchez, M.: Geodesic connectedness of semi-riemannian manifolds. Nonlinear Analy- sis: Theory, Methods & Applications 47(5), 3085–3102 (2001) https://doi.org/10.1016/ S0362-546X(01)00427-8 21

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Annali di Matem- atica Pura ed Applicata (1923 -) 200, 1135–1148 (2021) https://doi.org/10.1007/ s10231-020-01028-8 22

Silva, I., Flores, J.: Geodesic connectedness of affine manifolds. Annali di Matem- atica Pura ed Applicata (1923 -) 200, 1135–1148 (2021) https://doi.org/10.1007/ s10231-020-01028-8 22

work page 1923

[1] [1]

Mathematics 8(4) (2020) https://doi.org/10.3390/ math8040469

Al-Dayel, I., Deshmukh, S., Belova, O.: A remarkable property of concircular vec- tor fields on a riemannian manifold. Mathematics 8(4) (2020) https://doi.org/10.3390/ math8040469

work page 2020

[2] [2]

Amari, S.-i.: Information Geometry and Its Applications, p. 374. Springer, Tokyo (2018)

work page 2018

[3] [3]

Differential Geometry and its Applications 29(6), 779–786 (2011) https://doi.org/10.1016/j.difgeo.2011.08.006

Bartolo, R., Candela, A.M., Flores, J.L.: A note on geodesic connectedness of g ¨odel type spacetimes. Differential Geometry and its Applications 29(6), 779–786 (2011) https://doi.org/10.1016/j.difgeo.2011.08.006

work page doi:10.1016/j.difgeo.2011.08.006 2011

[4] [4]

Differential Geometry and its Applications8(3), 273–274 (1998) https://doi.org/10.1016/S0926-2245(98)00012-6

Bates, L.: You can’t get there from here. Differential Geometry and its Applications8(3), 273–274 (1998) https://doi.org/10.1016/S0926-2245(98)00012-6

work page doi:10.1016/s0926-2245(98)00012-6 1998

[5] [5]

Annali di Matematica Pura ed Applicata 155, 137–142 (1989) https://doi.org/10.1007/ BF01765938

Beem, J.K., Parker, P.E.: Pseudoconvexity and geodesic connectedness. Annali di Matematica Pura ed Applicata 155, 137–142 (1989) https://doi.org/10.1007/ BF01765938

work page 1989

[6] [6]

Hiroshima Mathematical Journal 23(2), 327–332 (1993) https://doi.org/10.32917/hmj/1206128255

Eguchi, S.: Geometry of minimum contrast. Hiroshima Mathematical Journal 22(3), 631–647 (1992) https://doi.org/10.32917/hmj/1206128508

work page doi:10.32917/hmj/1206128508 1992

[7] [7]

Information Geometry 4, 65–130 (2018) https://doi.org/10.1007/s41884-021-00047-5

Felice, D., Ay, N.: Towards a canonical divergence within information geometry. Information Geometry 4, 65–130 (2018) https://doi.org/10.1007/s41884-021-00047-5

work page doi:10.1007/s41884-021-00047-5 2018

[8] [8]

In: Geometry of Cauchy-Riemann Submanifolds, pp

Furuhata, H., Hasegawa, I.: Submanifold theory in holomorphic statistical manifolds. In: Geometry of Cauchy-Riemann Submanifolds, pp. 179–215. Springer, Singapore (2016). https://doi.org/10.1007/978-981-10-0916-7 7 20

work page doi:10.1007/978-981-10-0916-7 2016

[9] [9]

Information Geometry 5(1), 31–46 (2022) https://doi.org/10.1007/s41884-022-00065-x

Kobayashi, S., Ohno, Y.: On a constant curvature statistical manifold. Information Geometry 5(1), 31–46 (2022) https://doi.org/10.1007/s41884-022-00065-x

work page doi:10.1007/s41884-022-00065-x 2022

[10] [10]

Tohoku Mathematical Journal 46(3), 427–433 (1994) https://doi.org/10.2748/tmj/1178225722

Kurose, T.: On the divergences of 1-conformally flat statistical manifolds. Tohoku Mathematical Journal 46(3), 427–433 (1994) https://doi.org/10.2748/tmj/1178225722

work page doi:10.2748/tmj/1178225722 1994

[11] [11]

The Journal of Geometric Analysis 27, 2539–2556 (2016) https://doi.org/10.1007/s12220-017-9771-x

Li, J., Xia, C.: An integral formula for affine connections. The Journal of Geometric Analysis 27, 2539–2556 (2016) https://doi.org/10.1007/s12220-017-9771-x

work page doi:10.1007/s12220-017-9771-x 2016

[12] [12]

Hokkaido Mathematical Journal27(2), 409–421 (1998) https://doi.org/ 10.14492/HOKMJ/1351001292

Matsuzoe, H.: On realization of conformally-projectively flat statistical manifolds and the divergences. Hokkaido Mathematical Journal27(2), 409–421 (1998) https://doi.org/ 10.14492/HOKMJ/1351001292

work page doi:10.14492/hokmj/1351001292 1998

[13] [13]

In: Probabilistic Approach to Geometry vol

Matsuzoe, H.: Statistical manifolds and affine differential geometry. In: Probabilistic Approach to Geometry vol. 57, pp. 303–322. Mathematical Society of Japan, Tokyo (2010). https://doi.org/10.2969/aspm/05710303

work page doi:10.2969/aspm/05710303 2010

[14] [14]

Hiroshima Mathematical Journal 23(2), 327–332 (1993) https://doi.org/10.32917/hmj/1206128255

Matumoto, T.: Any statistical manifold has a contrast function—on the 𝐶3-functions taking the minimum at the diagonal of the product manifold. Hiroshima Mathematical Journal 23(2), 327–332 (1993) https://doi.org/10.32917/hmj/1206128255

work page doi:10.32917/hmj/1206128255 1993

[15] [15]

Revista de la Real Academia de Ciencias Exactas, F ´ısicas y Naturales

Meli, C.B., Ngakeu, F., Olea, B.: Statistical structures arising in null submanifolds. Revista de la Real Academia de Ciencias Exactas, F ´ısicas y Naturales. Serie A, Matem´aticas 117(1) (2023) https://doi.org/10.1007/s13398-022-01381-8

work page doi:10.1007/s13398-022-01381-8 2023

[16] [16]

Differential Geometry and its Applications 41, 39–47 (2015) https://doi.org/10.1016/j.difgeo.2015.04.005

Min, C., Ri, W., Kwak, K., An, D.: Equiaffine structure and conjugate ricci-symmetry of a statistical manifold. Differential Geometry and its Applications 41, 39–47 (2015) https://doi.org/10.1016/j.difgeo.2015.04.005

work page doi:10.1016/j.difgeo.2015.04.005 2015

[17] [17]

Differential Geometry and its Applications 2(3), 197–222 (1992) https://doi.org/10.1016/0926-2245(92)90011-B

Noguchi, M.: Geometry of statistical manifolds. Differential Geometry and its Applications 2(3), 197–222 (1992) https://doi.org/10.1016/0926-2245(92)90011-B

work page doi:10.1016/0926-2245(92)90011-b 1992

[18] [18]

Cambridge Tracts in Mathematics, vol

Nomizu, K., Sasaki, T.: Affine Differential Geometry. Cambridge Tracts in Mathematics, vol. 111, p. 263. Cambridge University Press, Cambridge (1994)

work page 1994

[19] [19]

Proceedings of the American Mathematical Society 12(6), 889–891 (1961)

Nomizu, K., Ozeki, H.: The existence of complete riemannian metrics. Proceedings of the American Mathematical Society 12(6), 889–891 (1961)

work page 1961

[20] [20]

O’Neill, B.: Semi-Riemannian Geometry With Applications to Relativity, 103, Volume 103 (Pure and Applied Mathematics) vol. 103. Academic Press, New York (1983)

work page 1983

[21] [21]

Annals of Global Analysis and Geometry 59(3), 367–383 (2021) https://doi.org/10.1007/s10455-021-09752-x

Opozda, B.: Completeness in affine and statistical geometry. Annals of Global Analysis and Geometry 59(3), 367–383 (2021) https://doi.org/10.1007/s10455-021-09752-x

work page doi:10.1007/s10455-021-09752-x 2021

[22] [22]

Nonlinear Analy- sis: Theory, Methods & Applications 47(5), 3085–3102 (2001) https://doi.org/10.1016/ S0362-546X(01)00427-8 21

S ´anchez, M.: Geodesic connectedness of semi-riemannian manifolds. Nonlinear Analy- sis: Theory, Methods & Applications 47(5), 3085–3102 (2001) https://doi.org/10.1016/ S0362-546X(01)00427-8 21

work page 2001

[23] [23]

Annali di Matem- atica Pura ed Applicata (1923 -) 200, 1135–1148 (2021) https://doi.org/10.1007/ s10231-020-01028-8 22

Silva, I., Flores, J.: Geodesic connectedness of affine manifolds. Annali di Matem- atica Pura ed Applicata (1923 -) 200, 1135–1148 (2021) https://doi.org/10.1007/ s10231-020-01028-8 22

work page 1923