On Nye’s Lattice Curvature Tensor

Representation of Nye’s Lattice Curvature Tensor by Log Angles

Journal of the Japan Institute of Metals and Materials ◽

10.2320/jinstmet.j2018033 ◽

2018 ◽

Vol 82 (11) ◽

pp. 415-418

Author(s):

Ryosuke Matsutani ◽

Susumu Onaka

Keyword(s):

Curvature Tensor ◽

Lattice Curvature

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Representation of Nye’s Lattice Curvature Tensor by Log Angles

MATERIALS TRANSACTIONS ◽

10.2320/matertrans.m2019049 ◽

2019 ◽

Vol 60 (6) ◽

pp. 935-938 ◽

Cited By ~ 1

Author(s):

Ryosuke Matsutani ◽

Susumu Onaka

Keyword(s):

Curvature Tensor ◽

Lattice Curvature

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Generalized Riemann spaces

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100031686 ◽

1956 ◽

Vol 52 (4) ◽

pp. 623-625 ◽

Cited By ~ 12

Author(s):

John Moffat

Keyword(s):

Curvature Tensor ◽

Physical Theory ◽

Dimensional Space ◽

Riemann Space ◽

Recent Attempt ◽

Fundamental Tensor ◽

Fundamental Properties ◽

Complex Symmetric ◽

Definition Of ◽

Generalized Curvature Tensor

ABSTRACTThe recent attempt at a physical interpretation of non-Riemannian spaces by Einstein (1, 2) has stimulated a study of these spaces (3–8). The usual definition of a non-Riemannian space is one of n dimensions with which is associated an asymmetric fundamental tensor, an asymmetric linear affine connexion and a generalized curvature tensor. We can also consider an n-dimensional space with which is associated a complex symmetric fundamental tensor, a complex symmetric affine connexion and a generalized curvature tensor based on these. Some aspects of this space can be compared with those of a Riemann space endowed with two metrics (9). In the following the fundamental properties of this non-Riemannian manifold will be developed, so that the relation between the geometry and physical theory may be studied.

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Curvature of quaternionic Kähler manifolds with $$S^1$$-symmetry

manuscripta mathematica ◽

10.1007/s00229-021-01294-7 ◽

2021 ◽

Author(s):

V. Cortés ◽

A. Saha ◽

D. Thung

Keyword(s):

Curvature Tensor ◽

Decomposition Theorem ◽

Kähler Manifolds ◽

Kahler Manifolds ◽

Riemann Curvature ◽

Riemann Curvature Tensor ◽

Cohomogeneity One ◽

Civita Connection ◽

Levi Civita Connection ◽

Quaternionic Kähler

AbstractWe study the behavior of connections and curvature under the HK/QK correspondence, proving simple formulae expressing the Levi-Civita connection and Riemann curvature tensor on the quaternionic Kähler side in terms of the initial hyper-Kähler data. Our curvature formula refines a well-known decomposition theorem due to Alekseevsky. As an application, we compute the norm of the curvature tensor for a series of complete quaternionic Kähler manifolds arising from flat hyper-Kähler manifolds. We use this to deduce that these manifolds are of cohomogeneity one.

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Analysis of the Curvature Tensor from the Viewpoint of Signal Processing

10.1063/1.2997300 ◽

2008 ◽

Author(s):

Lennart Wietzke ◽

Gerald Sommer ◽

Christian Schmaltz ◽

Joachim Weickert ◽

Theodore E. Simos ◽

...

Keyword(s):

Signal Processing ◽

Curvature Tensor

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Traceless component of the conformal curvature tensor in Kähler manifold

Czechoslovak Mathematical Journal ◽

10.1007/s10587-006-0061-1 ◽

2006 ◽

Vol 56 (3) ◽

pp. 857-874 ◽

Cited By ~ 2

Author(s):

S. Funabashi ◽

H. S. Kim ◽

Y.-M. Kim ◽

J. S. Pak

Keyword(s):

Curvature Tensor ◽

Kähler Manifold ◽

Conformal Curvature ◽

Conformal Curvature Tensor ◽

Kahler Manifold

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A Simple Algorithm to Eliminate Ambiguities in EBSD Orientation Map Visualization and Analyses: Application to Fatigue Crack-Tips/Wakes in Aluminum Alloys

Microscopy and Microanalysis ◽

10.1017/s1431927610093992 ◽

2010 ◽

Vol 16 (6) ◽

pp. 831-841 ◽

Cited By ~ 16

Author(s):

Vipul K. Gupta ◽

Sean R. Agnew

Keyword(s):

Fatigue Crack ◽

Aluminum Alloys ◽

Direct Determination ◽

Simple Algorithm ◽

Electron Backscattered Diffraction ◽

Lattice Curvature ◽

Orientation Data ◽

Orientation Map ◽

Crack Tips ◽

Map Visualization

AbstractA simple algorithm is developed and implemented to eliminate ambiguities, in both statistical analyses of orientation data (e.g., orientation averaging) and electron backscattered diffraction (EBSD) orientation map visualization, caused by symmetrically equivalent orientations and the wrap-around or umklapp effect. Using crystal symmetry operators and the lowest Euclidian-distance criterion, the orientation of each pixel within a grain is redefined. An advantage of this approach is demonstrated for direct determination of the representative orientation of a grain within an EBSD map by mean, median, or quaternion-based averaging methods that can be further used within analyses or visualization of misorientation or geometrically necessary dislocation (GND) density. If one also considers the lattice curvature tensor, five components of the dislocation density tensor—corresponding to a part of the GND content—may be inferred. The methodology developed is illustrated using EBSD orientation data obtained from the fatigue crack-tips/wakes in aerospace aluminum alloys 2024-T351 and 7050-T7451.

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