Tensors are multilinear geometric quantities with a prescribed coordinate transformation law. They include metrics, differential forms, curvature, and the linear operators used throughout geometry and physics.
Tensor spaces
Definition. A tensor of type \((r,s)\) at \(p\) is an element \[ T\in (T_pM)^{\otimes r}\otimes(T_p^*M)^{\otimes s}, \] equivalently a multilinear map with \(r\) covector and \(s\) vector inputs. A smooth tensor field is a smooth section of the corresponding tensor bundle.
In coordinates, \[ T=T^{i_1\cdots i_r}_{j_1\cdots j_s} \partial_{i_1}\otimes\cdots\otimes\partial_{i_r} \otimes\mathrm{d}x^{j_1}\otimes\cdots\otimes\mathrm{d}x^{j_s}. \] Under \(y=y(x)\), each upper index acquires a factor \(\partial y/\partial x\) and each lower index a factor \(\partial x/\partial y\). The transformation law follows from changing the vector and covector bases, not from an extra definition imposed on the components.
Tensor operations
Tensor products add types: \((r,s)\otimes(r',s')=(r+r',s+s')\). Contraction pairs one upper with one lower index and lowers both counts by one. For a \((1,1)\)-tensor \(A\), contraction is the coordinate-independent trace \(\operatorname{tr}A=A^i_i\).
Proposition. Contraction is well-defined independently of a basis.
Proof. For a decomposable tensor \(v\otimes\alpha\), contraction is \(\alpha(v)\), an invariant pairing. Extend linearly and tensor with the remaining factors. Equivalently, in a coordinate transformation the paired Jacobian matrices cancel by the chain rule.
Pullback and pushforward
If \(F:M\to N\), covariant tensors pull back: \[ (F^*T)_p(v_1,\ldots,v_s) =T_{F(p)}(\mathrm{d}F_pv_1,\ldots,\mathrm{d}F_pv_s). \] Contravariant tensors generally cannot be pushed forward unless \(F\) is a diffeomorphism or additional compatibility is available. This asymmetry explains why differential forms behave especially well under arbitrary smooth maps.
Proposition. Pullback is functorial: \[ (G\circ F)^*=F^*\circ G^*,\qquad \operatorname{id}^*=\operatorname{id}. \]
Proof. Insert \(\mathrm{d}(G\circ F)=\mathrm{d}G\circ\mathrm{d}F\) into each argument of the covariant tensor.
Symmetry and alternating tensors
A covariant \(k\)-tensor is symmetric if permutations leave it unchanged, and alternating if they multiply it by the sign. Symmetrization and alternation are projections: \[ \operatorname{Sym}T=\frac1{k!}\sum_{\sigma\in S_k}\sigma T,\qquad \operatorname{Alt}T=\frac1{k!}\sum_{\sigma\in S_k} \operatorname{sgn}(\sigma)\sigma T. \] Riemannian metrics are positive-definite symmetric \(2\)-tensors. Differential \(k\)-forms are alternating covariant \(k\)-tensors.
Why ordinary derivatives of components are not tensors
Example. If \(X=X^i\partial_i\), then under a coordinate change the derivatives \(\partial_jX^i\) acquire second derivatives of the transition map. They do not transform tensorially. A connection adds correction coefficients so that \(\nabla_jX^i=\partial_jX^i+\Gamma^i_{jk}X^k\) does transform as a \((1,1)\)-tensor. Geometry therefore distinguishes intrinsic tensors from coordinate arrays that only resemble them.