If $A^iB_i$ is called a contraction, what is $A^{ij}B_{ij}$ called?












0














I have a tensor $A^{ijk}$ and a tensor $B_{ijk}$, and I'd like to contract all the indices between them, resulting in the scalar $k$:



$$k = A^{ijk}B_{ijk}$$



Is there a name for this sort of "multi-contraction"?



By knowing the name, I'd like to study this type of operation more. I'm not sure whether they occur in the study of differential forms, or tensor algebra, or geometric algebra, or otherwise.






reference-request tensors geometric-algebras







share|cite|improve this question


















  • 1




    I believe this one is also called just contraction.
    – lisyarus
    Nov 27 '18 at 21:25










  • lisyarus is right.
    – J.G.
    Nov 27 '18 at 21:41
















0














I have a tensor $A^{ijk}$ and a tensor $B_{ijk}$, and I'd like to contract all the indices between them, resulting in the scalar $k$:



$$k = A^{ijk}B_{ijk}$$



Is there a name for this sort of "multi-contraction"?



By knowing the name, I'd like to study this type of operation more. I'm not sure whether they occur in the study of differential forms, or tensor algebra, or geometric algebra, or otherwise.






reference-request tensors geometric-algebras







share|cite|improve this question


















  • 1




    I believe this one is also called just contraction.
    – lisyarus
    Nov 27 '18 at 21:25










  • lisyarus is right.
    – J.G.
    Nov 27 '18 at 21:41














0












0








0







I have a tensor $A^{ijk}$ and a tensor $B_{ijk}$, and I'd like to contract all the indices between them, resulting in the scalar $k$:



$$k = A^{ijk}B_{ijk}$$



Is there a name for this sort of "multi-contraction"?



By knowing the name, I'd like to study this type of operation more. I'm not sure whether they occur in the study of differential forms, or tensor algebra, or geometric algebra, or otherwise.






reference-request tensors geometric-algebras







share|cite|improve this question













I have a tensor $A^{ijk}$ and a tensor $B_{ijk}$, and I'd like to contract all the indices between them, resulting in the scalar $k$:



$$k = A^{ijk}B_{ijk}$$



Is there a name for this sort of "multi-contraction"?



By knowing the name, I'd like to study this type of operation more. I'm not sure whether they occur in the study of differential forms, or tensor algebra, or geometric algebra, or otherwise.






reference-request tensors geometric-algebras




reference-request tensors geometric-algebras






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Nov 27 '18 at 21:22









Doubt

738321




738321








  • 1




    I believe this one is also called just contraction.
    – lisyarus
    Nov 27 '18 at 21:25










  • lisyarus is right.
    – J.G.
    Nov 27 '18 at 21:41














  • 1




    I believe this one is also called just contraction.
    – lisyarus
    Nov 27 '18 at 21:25










  • lisyarus is right.
    – J.G.
    Nov 27 '18 at 21:41








1




1




I believe this one is also called just contraction.
– lisyarus
Nov 27 '18 at 21:25




I believe this one is also called just contraction.
– lisyarus
Nov 27 '18 at 21:25












lisyarus is right.
– J.G.
Nov 27 '18 at 21:41




lisyarus is right.
– J.G.
Nov 27 '18 at 21:41










1 Answer
1






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oldest

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0














They occur in many places. An example is the scalar curvature



$$
S = g^{ij}R_{ij}
$$



where $g^{ij}$ is the metric tensor and $R^{ij}$ the Ricci curvature, which in turn is also the contraction of the Riemann curvature tensor






share|cite|improve this answer

















  • 1




    Some authors also use the term "trace" or "metric trace" in your example. I personally use contraction when dealing with the operation on mixed tensors, e.g., Ricci curvature is the contraction of the Riemannian curvature, $R_{ij}=R_{kij}^{,,,k}$; and trace to indicate metric dependence, e.g., scalar curvature is the trace of the Ricci curvature, $S=text{tr}_g(text{Ric})=g^{ij}R_{ij}$.
    – Matt
    Nov 28 '18 at 9:11











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1 Answer
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1 Answer
1






active

oldest

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active

oldest

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active

oldest

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0














They occur in many places. An example is the scalar curvature



$$
S = g^{ij}R_{ij}
$$



where $g^{ij}$ is the metric tensor and $R^{ij}$ the Ricci curvature, which in turn is also the contraction of the Riemann curvature tensor






share|cite|improve this answer

















  • 1




    Some authors also use the term "trace" or "metric trace" in your example. I personally use contraction when dealing with the operation on mixed tensors, e.g., Ricci curvature is the contraction of the Riemannian curvature, $R_{ij}=R_{kij}^{,,,k}$; and trace to indicate metric dependence, e.g., scalar curvature is the trace of the Ricci curvature, $S=text{tr}_g(text{Ric})=g^{ij}R_{ij}$.
    – Matt
    Nov 28 '18 at 9:11
















0














They occur in many places. An example is the scalar curvature



$$
S = g^{ij}R_{ij}
$$



where $g^{ij}$ is the metric tensor and $R^{ij}$ the Ricci curvature, which in turn is also the contraction of the Riemann curvature tensor






share|cite|improve this answer

















  • 1




    Some authors also use the term "trace" or "metric trace" in your example. I personally use contraction when dealing with the operation on mixed tensors, e.g., Ricci curvature is the contraction of the Riemannian curvature, $R_{ij}=R_{kij}^{,,,k}$; and trace to indicate metric dependence, e.g., scalar curvature is the trace of the Ricci curvature, $S=text{tr}_g(text{Ric})=g^{ij}R_{ij}$.
    – Matt
    Nov 28 '18 at 9:11














0












0








0






They occur in many places. An example is the scalar curvature



$$
S = g^{ij}R_{ij}
$$



where $g^{ij}$ is the metric tensor and $R^{ij}$ the Ricci curvature, which in turn is also the contraction of the Riemann curvature tensor






share|cite|improve this answer












They occur in many places. An example is the scalar curvature



$$
S = g^{ij}R_{ij}
$$



where $g^{ij}$ is the metric tensor and $R^{ij}$ the Ricci curvature, which in turn is also the contraction of the Riemann curvature tensor







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Nov 27 '18 at 21:31









caverac

13.8k21030




13.8k21030








  • 1




    Some authors also use the term "trace" or "metric trace" in your example. I personally use contraction when dealing with the operation on mixed tensors, e.g., Ricci curvature is the contraction of the Riemannian curvature, $R_{ij}=R_{kij}^{,,,k}$; and trace to indicate metric dependence, e.g., scalar curvature is the trace of the Ricci curvature, $S=text{tr}_g(text{Ric})=g^{ij}R_{ij}$.
    – Matt
    Nov 28 '18 at 9:11














  • 1




    Some authors also use the term "trace" or "metric trace" in your example. I personally use contraction when dealing with the operation on mixed tensors, e.g., Ricci curvature is the contraction of the Riemannian curvature, $R_{ij}=R_{kij}^{,,,k}$; and trace to indicate metric dependence, e.g., scalar curvature is the trace of the Ricci curvature, $S=text{tr}_g(text{Ric})=g^{ij}R_{ij}$.
    – Matt
    Nov 28 '18 at 9:11








1




1




Some authors also use the term "trace" or "metric trace" in your example. I personally use contraction when dealing with the operation on mixed tensors, e.g., Ricci curvature is the contraction of the Riemannian curvature, $R_{ij}=R_{kij}^{,,,k}$; and trace to indicate metric dependence, e.g., scalar curvature is the trace of the Ricci curvature, $S=text{tr}_g(text{Ric})=g^{ij}R_{ij}$.
– Matt
Nov 28 '18 at 9:11




Some authors also use the term "trace" or "metric trace" in your example. I personally use contraction when dealing with the operation on mixed tensors, e.g., Ricci curvature is the contraction of the Riemannian curvature, $R_{ij}=R_{kij}^{,,,k}$; and trace to indicate metric dependence, e.g., scalar curvature is the trace of the Ricci curvature, $S=text{tr}_g(text{Ric})=g^{ij}R_{ij}$.
– Matt
Nov 28 '18 at 9:11


















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